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The goal of this clinical research study is to learn if MEDI4736 given in combination with selumetinib can help to control advanced lung cancer.

The safety of this drug combination will also be studied.

Study Groups:

If participant is found to be eligible to take part in this study, participant will be randomly assigned (as in the flip of a coin) to 1 of 2 study groups. This is done because no one knows if one study group is better, the same, or worse than the other group.

• If participant is in Group 1, participant will take selumetinib 2 times each day.
• If participant is in Group 2, participant will take selumetinib 2 times each day on Days 1-7 and 15-22 of each study cycle.

All participants will also receive MEDI4736.

Study Drug Administration:

Each study cycle is 28 days.

Participant should take each dose of selumetinib about 12 hours apart after fasting (having nothing to eat or drink except water) for 1 hour before and 2 hours after taking selumetinib. Participant will follow the dosing schedule to which participant has been assigned (as described above under Study Groups).

Participant should record each selumetinib dose in the dosing diary that will be given to participant. Participant should bring the diary to participant's study visits at the end of every cycle.

Both groups will receive MEDI4736 by vein over about 60 minutes on Day 1 of every cycle.

Participant will be given standard drugs to help decrease the risk of side effects. Participant may ask the study staff for information about how the drugs are given and their risks.

Length of Study:

Participant may receive the study drugs for as long as the doctor thinks is in participant's best interest. Participant will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if participant is unable to follow study directions.

Participation on the study will be over after the Follow-Up Period.

Study Visits:

About 14 days before the first dose of the study drugs:

• Participant will have a physical exam.
• Blood (about 3 teaspoons) and urine will be collected for routine tests.
• If the doctor thinks it is needed, participant will have an EKG.

On Day 1 of every cycle:

• Participant will have a physical exam.
• Blood (about 3 teaspoons) and urine will be collected for routine tests.
• If the doctor thinks it is needed, participant will have an eye exam.
• If participant can become pregnant and the doctor thinks it is needed, blood (about ½ teaspoon) or urine will be collected for a pregnancy test.

On Day 1 of Cycle 1 and at Week 16, participant will have an EKG. Participant will have triplicate EKGs on Day 1 of Cycle 1.

At Week 8 and about every 8 weeks after that, participant will have a CT scan or MRI to check the status of the disease.

At Week 12 and about every 12 weeks after that, participant will have an ECHO or multigated acquisition (MUGA) scan.

End-of-Study Visits:

At about 30 days and 90 days after participant's last dose of the study drugs:

• Participant will have a physical exam.
• Blood (about 3 teaspoons) will be drawn for routine tests.
• Participant will have an EKG.
• If participant can become pregnant and the doctor thinks it is needed, blood (about ½ teaspoon) or urine will be collected for a pregnancy test.

Follow Up Period:

About every 6 months, participant will be called by a member of the study staff to ask how participant is doing and if participant has started any new treatments outside of the study. These calls should last about 10 minutes each time.

If the doctor thinks it is needed, participant may return to the clinic for either a CT scan or MRI to check the status of the disease.

This is an investigational study. MEDI4736 and selumetinib are not FDA approved or commercially available. They are currently being used for research purposes only. The study doctor can explain how the study drugs are designed to work.

Up to 76 participants will be enrolled in this study. All will take part at MD Anderson.

Inclusion Criteria:

1. Histologically or cytologically confirmed recurrent non-small cell lung cancer not amendable to curative intent therapy or stage IV NSCLC.
2. Known KRAS mutation status by CLIA certified test. Patients in the safety run-in are not required to have a tumor with mutant KRAS. In the randomized portion of the trial, only patients with KRAS mutation are eligible.
3. Documented progression following at least one line of chemotherapy for metastatic or recurrent disease, or progression within 6 months of receiving adjuvant chemotherapy or concurrent chemotherapy for early stage or locally advanced disease.
4. Biopsy accessible disease and willingness to undergo tumor biopsy.
5. Measurable disease by RECIST 1.1.
6. Age>/= 18 years.
7. ECOG performance status 0 or 1.
8. Ability to take pills by mouth.
9. Patients must have normal organ and marrow function as defined: leukocytes >/=3,000/mcL, absolute neutrophil count >/=1,500/mcL, platelets >/=100,000/mcL, hemoglobin >/9.0g/dL, total bilirubin </=1.5 x upper limit of normal (ULN) (higher is allowed if in the setting of known Gilbert's disease), AST(SGOT)/ALT(SGPT) </=2.5 x institutional upper limit of normal or </=5 x ULN if liver metastases are present, Alkaline phosphatase </=3.5 x institutional upper limit of normal or <6 x ULN if liver metastases are present, creatinine clearance >/=50 mL/min/1.73^2 by Cockcroft-Gault equation (creatinine clearance= ([140-age]x body mass)/(plasma creatinine x 72) x gender correction factor) or by 24-hour urine collection.
10. Brain metastases are allowed, as long as they are stable and do not require treatment with anticonvulsants or escalating doses of steroids.
11. Females of childbearing potential must have a negative serum pregnancy test and must agree to use adequate contraception for the duration of the study and six months after. Females of childbearing potential are defined as those who are not surgically sterile (ie, bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or postmenopausal (defined as 12 months with no menses without an alternative medical cause). Continued in #12
12. Continued from #11: a) Acceptable effective methods of contraception for women include implants, injectables, combined oral contraceptives, some intrauterine devices/systems and sterilization including vasectomy of the partner, all being used in combination with barrier methods of contraception (e.g. condoms). b) True sexual abstinence is also an acceptable method of contraception. Continued in #13
13. Continued from #12: Women will be considered post-menopausal if they have been amenorrheic for the past 12 months without an alternative medical cause. The following age-specific requirements must also apply: i.)Women < 50 years old: they would be considered post-menopausal if they have been amenorrheic for the past 12 months or more following cessation of exogenous hormonal treatments. The levels of Luteinizing Hormone (LH) and Follicle-Stimulating Hormone (FSH) must also be in the post-menopausal range (as per the institution). ii.)Women >/= 50 years old: they would be consider post-menopausal if they have been amenorrheic for the past 12 months or more following cessation of all exogenous hormonal treatments, or have had radiation-induced oophorectomy with the last menses > 1 year ago, or have had chemotherapy-induced menopause with >1 year interval since last menses, or have had surgical sterilisation by either bilateral oophorectomy or hysterectomy.
14. Non-sterilized males who are sexually active with a female partner of childbearing potential must use adequate contraception for the duration of the study and 90 days after the last dose of study medication. a) Acceptable methods of contraception for men include the use of condoms with spermicidal foams/gels or prior vasectomy. b) True sexual abstinence is also an acceptable method of contraception.
15. Ability to understand and the willingness to sign a written informed consent document.
16. Have adequate renal function, with a GFR of >/= 50ml/min by the Cockcroft-Gault formula or by 24 hour urine collection.

Exclusion Criteria:

1. Have received or are receiving an investigational medicinal product (IMP) or other systemic anticancer treatment within 4 weeks prior to the first dose of study treatment, or within a period during which the IMP or systemic anticancer treatment has not been cleared from the body (e.g. a period of 5 'half-lives'), whichever is the most appropriate and as judged by the investigator.
2. Current or prior use of immunosuppressive medication within 14 days of the 1st dose of durvalumab, with the exception of intranasal and inhaled corticosteroids or oral corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone, or an equivalent corticosteroid.
3. Any concurrent chemotherapy, immunotherapy, biologic, or hormonal therapy for cancer treatment.
4. Receipt of radiation therapy within 4 weeks prior to starting study treatment. Limited field of radiation for palliation at any time prior to the start of study treatment is acceptable if: i.) The lung is not in the radiation field ii.) The irradiated lesions are not used as target lesions.
5. Receipt of live attenuated vaccination within 30 days prior to study entry or within 30 days of receiving durvalumab.
6. Prior treatment with a MEK, Ras, or Raf inhibitor, or CTLA4 inhibitor.
7. Patients who are receiving any other investigational agents.
8. Have any unresolved chronic toxicity with CTCAE grade >/= 2, from previous anticancer therapy, except for alopecia.
9. Known hypersensitivity to selumetinib or durvalumab or any excipient or history of allergic reactions attributed to compounds of similar chemical or biologic composition to selumetinib or durvalumab.
10. Active or prior documented autoimmune disease within the past 3 years. Patients with a history of vitiligo, Grave's disease, or psoriasis not requiring systemic treatment (within the past 2 years) are not excluded.
11. Active or prior documented inflammatory bowel disease (e.g. Crohn's disease, ulcerative colitis).
12. Have known or suspected brain metastases or spinal cord compression, unless the condition has been asymptomatic, has been treated with surgery and / or radiation, and has been stable without requiring corticosteroids nor anti-convulsant medications for at least 4 weeks prior to the first dose of study medication.
13. Known history of previous clinical diagnosis of tuberculosis.
14. History of primary immunodeficiency.
15. History of organ transplant requiring therapeutic immunosuppression.
16. Cardiac conditions as follows: i.) Mean QT interval corrected for heart rate (QTc) >/=450 ms calculated from 3 ECGs using Fredericia's formula [QTcF] or other factors that increase the risk of QT prolongation. ii.) Uncontrolled hypertension (BP >/= 150/95 despite optimal medical therapy). iii.) Acute coronary syndrome within 6 months prior to starting treatment. iv.) Uncontrolled Angina - Canadian Cardiovascular Society grade II-IV despite medical therapy. v.) Symptomatic heart failure NYHA Class II-IV, prior or current cardiomyopathy, or severe valvular heart disease. - Criteria continued in #17
17. Continued from #16: vi.) Prior or current cardiomyopathy including but not limited to the following: a) known hypertrophic cardiomyopathy b) known arrhythmogenic right ventricular cardiomyopathy c) previous moderate or severe impairment of left ventricular systolic function (LVEF <45% on echocardiography or equivalent on MuGA) even if full recovery has occurred. vii.) Baseline Left ventricular ejection fraction (LVEF) below the LLN or <55% measured by echocardiography or institution's LLN for MUGA. viii.) Severe valvular heart disease. ix.) Atrial fibrillation with a ventricular rate > 100 bpm on ECG at rest.
18. Ophthalmologic conditions as follows: i.) Current or past history of retinal pigment epithelial detachment (RPED)/central serous retinopathy (CSR) or retinal vein occlusion. ii.) Intraocular pressure (IOP) > 21 mmHg or uncontrolled glaucoma (irrespective of IOP).
19. Any gastrointestinal disorder expected to limit absorption of selumetinib
20. History of another primary malignancy within 5 years prior to starting study treatment, except for adequately treated basal or squamous cell carcinoma of the skin or cancer of the cervix in situ
21. Recent major surgery within 4 weeks prior to starting study treatment, with the exception of surgical placement for vascular access.
22. Uncontrolled intercurrent illness including, but not limited to, uncompensated respiratory, cardiac, hepatic, or renal disease, active infection (including hepatitis B, hepatitis C, HIV, and active clinical tuberculosis), active bleeding diatheses or renal transplant; ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, or psychiatric illness/social situations that would limit compliance with study requirements.
23. Pregnant or breastfeeding women.
24. Receiving or have received systemic anti-cancer therapy within 4 weeks prior to starting study treatment (6 weeks for nitrosoureas, mitomycin, and suramin), or any anticancer therapy which has not been cleared from the body by the time of starting study treatment.
25. Have evidence of any other significant clinical disorder or laboratory finding that, as judged by the investigator, makes it undesirable for the patient to participate in the study.
26. Have refractory nausea and vomiting, chronic gastrointestinal diseases (e.g., inflammatory bowel disease), or significant bowel resection that would adversely affect the absorption / bioavailability of the orally administered study medication.
27. Are male or female patients of reproductive potential and, as judged by the investigator, are not employing an effective method of birth control.
28. Patient weight less than 30 kg.