The goal of this clinical research study is to learn about the safety of giving immune cells called natural killer (NK) cells with chemotherapy to patients with leukemia, lymphoma, or multiple myeloma.

Immune system cells (such as NK cells) are made by the body to attack foreign or cancerous cells. Researchers think that NK cells you receive from a donor may react against cancer cells in your body, which may help to control the disease.

Primary Objective:

To determine the safety, efficacy and optimal cell dose of CAR.70/IL15-transduced CB-NK cells in patients with relapsed/refractory hematological malignances. The efficacy and optimal dose will be identified for individual diseases.

Secondary Objectives:

• To quantify persistence of infused allogeneic donor CAR-transduced CB-derived NK cells in the recipient.
• To conduct comprehensive immune reconstitution studies.

Inclusion criteria:

1. Patients with hematological malignances with an expression of CD70 in the pre-enrollment tumor sample ≥ 10% measured by immunohistochemistry or flow cytometry.

2. Patients must meet diseases specific eligibility criteria (see below)

3. Patients at least 1 week from last cytotoxic chemotherapy at the time of starting lymphodepleting chemotherapy, except for Hydroxyurea which is allowed for peripheral blood count control in AML, CML, and MDS patients until the day prior to administration of lymphodepleting chemotherapy. Patients may continue tyrosine kinase inhibitors or other targeted therapies until up to three days prior to administration of lymphodepleting chemotherapy.

4. Localized radiotherapy to one or more disease sites is allowed prior the infusion provided that there are additional disease sites that are not irradiated to assess response

5. Karnofsky Performance Scale > 50% for patients who are >16 years old or Lansky score ≥50% for patients who are ≤16 years of age.

6. Adequate organ function:

   1. Renal: Serum creatinine </= 2x ULN or estimated Glomerular Filtration Rate >/= 30 ml/min/1.73 m2
   2. Hepatic: ALT/AST </= 3 x ULN or </= 5 x ULN if documented liver metastases, Total bilirubin </2xULN, except in subjects with Gilbert's Syndrome in whom total bilirubin must be </= 3 x.ULN. No history of liver cirrhosis. No ascites.
   3. Cardiac: Cardiac ejection fraction >/= 40%, no clinically significant pericardial effusion as determined by an ECHO, and no uncontrolled arrhythmias or symptomatic cardiac disease.
   4. Pulmonary: No clinically significant pleural effusion (per PI discretion), baseline oxygen saturation > 92% on room air and adequate pulmonary function with FEV1, FVC and DLCO (corrected for Hgb) >50%.

7. Able to provide written informed consent.

8. 12-80 years of age.

9. Weight ≥40 kg

10. All participants who are able to have children must practice effective birth control while on study and up to 3 months post completion of study therapy. Acceptable forms of birth control for female patients include: hormonal birth control, intrauterine device, diaphragm with spermicide, condom with spermicide, or abstinence, for the length of the study. If the participant is a female and becomes pregnant or suspects pregnancy, she must immediately notify her doctor. If the participant becomes pregnant during this study, she will be taken off this study. Men who are able to have children must use effective birth control while on the study. If the male participant fathers a child or suspects that he has fathered a child while on the study, he must immediately notify his doctor.

11. Signed consent to long-term follow-up protocol PA17-0483 to fulfill the institutional responsibilities to various regulatory agencies.

12. Are willing and able to provide informed consent, as appropriate (either directly or through a legally authorized representative [LAR])

Exclusion criteria:

1. Positive beta HCG in female of child-bearing potential defined as not postmenopausal for 24 months or no previous surgical sterilization or lactating females.
2. Presence of clinically significant Grade 3 or greater toxicity from the previous treatment, as determined by PI.
3. Presence of uncontrolled fungal, bacterial, viral, or other infection not responding to appropriate therapy.
4. HIV with detectable viral load
5. Presence of active neurological disorder(s).
6. Active autoimmune disease within 12 months of enrollment
7. Amyloidosis or POEMS syndrome
8. Active cerebral or meningeal involvement by the malignancy
9. Active (defined as requiring therapy) acute or chronic GVHD
10. Any other malignancy known to be active, except for treated cervical intra-epithelial neoplasia and non-melanoma skin cancer.
11. Presence of any other serious medical condition that may endanger the patient at investigator discretion.
12. Major surgery <4 weeks prior to first dose of the preparatory chemotherapy
13. Allogeneic SCT or DLI <12 weeks prior to first dose of preparatory chemotherapy
14. Concomitant use of other investigational agents.
15. Concomitant use of other anti-cancer agents.
16. Patients receiving systemic steroid therapy at time of NK cell infusion (physiological substitutive doses are allowed), or have received antithymocyte globulin or lymphocyte immune globulin within 14 days of enrollment or alemtuzumab within 28 days of enrollment.
17. Patients receiving immunosuppressive therapy