Change in treatment priorities for these disease types as well as staffing concerns.

The goal of this clinical research study is to find the highest tolerable dose of TAK228 that can be given in combination with brentuximab vedotin in patients with lymphoma. The safety of this combination will also be studied.

This is an investigational study. TAK228 is not FDA approved or commercially available. It is currently being used for research purposes only. Brentuximab vedotin is FDA approved and commercially available for the treatment of different types of lymphoma. The study doctor can explain how the study drugs are designed to work.

Up to 18 participants will be enrolled in this study. All will take part at MD Anderson.

Study Groups:

If you are found to be eligible to take part in this study, you will be assigned to a dose level of TAK228 based on when you join this study. Up to 4 dose levels of TAK228 will be tested. About 3-6 participants will be enrolled at each dose level. The first group of participants will receive the lowest dose level. Each new group will receive a higher dose than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of TAK228 is found.

Your dose of brentuximab vedotin will stay the same. However, if the doctor thinks it is needed for your safety, it may be lowered.

Study Drug Administration:

Each cycle is 21 days.

You will receive brentuximab vedotin by vein over about 30 minutes on Day 1 of each cycle.

You will take TAK228 by mouth either every day or on a 5 days on/2 days off schedule (a "5/2 schedule," where you take the study drug every day for 5 days in a row and then do not take any TAK228 for 2 days in a row). The study doctor will tell you how often to take the study drug.

Each dose of TAK228 should be taken at about the same time each day on an empty stomach with 8 ounces (about 1 cup) of water. You should fast for 2 hours before and 1 hour after each dose.

Length of Study:

You may receive up to 16 cycles of study drugs. You will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if you are unable to follow study directions.

Your participation in this study will be over after about 2 years of follow-up (described below).

Study Visits:

One (1) time every week while receiving the study drugs, blood (about 1-2 tablespoons) will be drawn for routine tests.

Within 3 days before the start of Cycles 2 and beyond:

• You will have a physical exam.
• Blood (about 1-2 tablespoons) will be drawn for routine tests and to check your blood sugar levels. If you can become pregnant, part of this blood sample will be used for a pregnancy test. You should fast for about 8-10 hours before this draw.
• Urine will be collected for routine tests.
• You will have an EKG.

At the end of Cycle 1:

°Blood (about 1-2 tablespoons) will be drawn for biomarker testing.

At the end of Cycle 3 and every 3 cycles after that (Cycles 6, 9, 12, and so on), you will have a CT or PET/CT scan.

At-Home Glucose Monitoring:

You are required to monitor your glucose (sugar) levels at home during the first 2 months you are taking the study drug. If the doctor thinks it is needed, you may be asked to continue monitoring your glucose (sugar) levels at home. The study staff will give you a glucose monitor (called a glucometer) and teach you how and when to use it. You will bring the glucometer with you to each study visit so the study staff can collect the results of the testing.

The study team will tell you what an "abnormal" level is and when to contact the study doctor/study staff.

You will need to return the glucometer to the study staff at the end of the study.

End-of-Dosing Visit:

Within 21 days after your last dose of study drugs:

• You will have a physical exam.
• Blood (about 1-2 tablespoons) will be drawn for routine tests and to check your blood sugar levels. If you can become pregnant, part of this sample will be used for a pregnancy test. You should fast for about 8-10 hours before this draw.
• Urine will be collected for routine tests.
• You will have an EKG.
• You will have a PET/CT or CT scan.

Follow-Up:

Every 4 months for up to 2 years:

• You will have a physical exam.
• Blood (about 1-2 tablespoons) will be drawn for routine tests and to check your blood sugar levels. You should fast for about 8-10 hours before this blood draw.
• You will have a CT scan.

If you are found to be eligible to receive a transplant as part of your standard care, you may receive a transplant. You will sign a separate consent explaining that procedure and its risks in more detail. If you have a stem cell transplant, you will stop having follow-up visits as part of this study.

Inclusion Criteria:

1. Male or female patients 18 years or older.
2. Patients must have a diagnosis of relapsed or refractory classical Hodgkin lymphoma OR anaplastic large cell lymphoma OR non-ALCL peripheral T-cell lymphoma with a pre-enrollment tumor biopsy positive for CD30 by immunohistochemistry at >/= 1%. All patients must be refractory to or not eligible for available therapies expected to provide clinical benefit with the exception that if a patient would meet National Comprehensive Cancer Center Network (NCCN) guidelines for consideration of treatment with brentuximab vedotin monotherapy then that patient can be enrolled to this trial as this is a combination trial of brentuximab vedotin plus TAK228.
3. Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status </= 2
4. For women: Postmenopausal for at least 1 year before the screening visit, OR surgically sterile, OR if they are of childbearing potential, agree to practice 1 effective methods of contraception, and 1 additional effective (barrier) method, at the same time, from the time of signing the informed consent through 180 days (or longer, as mandated by local labeling [eg, USPI, SmPC, etc.])after the last dose of study drug, OR agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the patient (Periodic abstinence [e.g, calendar, ovulation, symptothermal, postovulation methods] and withdrawal, spermicide only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.)
5. #4 continued: For men, even if surgically sterilized (ie, status post-vasectomy), they must: Agree to practice highly effective barrier contraception during the entire study treatment period and through 180 days after the last dose of study drug, OR agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the patient (Periodic abstinence [e.g, calendar, ovulation, symptothermal, postovulation methods for the female partner] and withdrawal, spermicide only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.). Agree not to donate sperm during the course of this study or 180 days after receiving their last dose of study drug
6. Screening clinical laboratory values as specified below: a) Bone marrow reserve consistent with: absolute neutrophil count (ANC) >/= 1 x 10^9/L; platelet count >/= 90 x 10^9/L (for patients with bone marrow involvement a platelet count of >/= 75 x 10^9/L if needed) ; hemoglobin >/= 8 g/dL without transfusion within 1 week preceding study drug administration b) Hepatic: total bilirubin </= 1.5 x upper limit of normal (ULN), transaminases (aspartate aminotransferase/serum glutamic oxaloacetic transaminase-AST/SGOT and alanine aminotransferase/serum glutamic pyruvic transaminase-ALT/SGPT) </= 2.5 x ULN (</= 5 x ULN if liver metastases are present); c) Renal: creatinine clearance >/=50 mL/min based either on Cockcroft-Gault estimate or based on urine collection (12 or 24 hour); d) Metabolic: Glycosylated hemoglobin (HbA1c)<7.0%, fasting serum glucose (</=130 mg/dL) and fasting triglycerides </= 300 mg/dL;
7. Ability to swallow oral medications.
8. Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care.
9. Bi-dimensionally measurable disease with at least 1 lesion >/= 1.5 cm in a single dimension.

Exclusion Criteria:

1. Central nervous system (CNS) metastasis.
2. Other clinically significant co-morbidities, such as uncontrolled pulmonary disease, active central nervous system disease, active infection, or any other condition that could compromise the patient's participation in the study.
3. Known human immunodeficiency virus infection.
4. Known hepatitis B surface antigen-positive, or known or suspected active hepatitis C infection.
5. Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol.
6. Diagnosed or treated for another malignancy within 2 years before administration of the first dose of study drug, or previously diagnosed with another malignancy and have any evidence of residual disease. Patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection.
7. Breast feeding or pregnant.
8. Poorly controlled diabetes mellitus defined as glycosylated hemoglobin (HbA1c) > 7%; patients with a history of transient glucose intolerance due to corticosteroid administration may be enrolled in this study if all other inclusion/exclusion criteria are met
9. Treatment with strong inhibitors and/or inducers of cytochrome P450 (CYP) 3A4, CYP2C9 or CYP2C19 within 1 week preceding the first dose of study drug
10. Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI disease, or for an unknown reason that may alter the absorption of TAK228. In addition, patients with enteric stomata are also excluded.
11. Treatment with any investigational products within 14 days before the first dose of study drug.
12. History of any of the following within the last 6 months before administration of the first dose of the drug: Ischemic myocardial event, including angina requiring therapy and artery revascularization procedures. Ischemic cerebrovascular event, including transient ischemic attack and artery revascularization procedures. Requirement for inotropic support (excluding digoxin) or serious (uncontrolled) cardiac arrhythmia (including atrial flutter/fibrillation, ventricular fibrillation or ventricular tachycardia). Placement of a pacemaker for control of rhythm. New York Heart Association (NYHA) Class III or IV heart failure. Pulmonary embolism.
13. Significant active cardiovascular or pulmonary disease including: Uncontrolled hypertension (i.e., systolic blood pressure >180 mm Hg, diastolic blood pressure > 95 mm Hg). Use of anti-hypertensive agents to control hypertension before Cycle1 Day 1 is allowed. Pulmonary hypertension. Uncontrolled asthma or O2 saturation < 90% by arterial blood gas analysis or pulse oximetry on room air. Significant valvular disease; severe regurgitation or stenosis by imaging independent of symptom control with medical intervention, or history of valve replacement. Medically significant (symptomatic) bradycardia. History of arrhythmia requiring an implantable cardiac defibrillator. Baseline prolongation of the rate-corrected QT interval (QTc) (e.g., repeated demonstration of QTc interval > 480 milliseconds, or history of congenital long QT syndrome, or torsades de pointes).
14. Patients receiving systemic corticosteroids (either intravenous [IV] or oral steroids, excluding inhalers or low-dose hormone replacement therapy) within 1 week before administration of the first dose of study drug.
15. Daily or chronic use of a proton pump inhibitor (PPI) and/or having taken a PPI within 7 days before receiving the first dose of study drug.
16. Patients who have undergone past allogeneic stem cell transplant must be 1 year from completion of transplant and have not active graft versus host disease