INCLUSION CRITERIA

Subjects must meet all of the following criteria to be included in this study:

1. Signed written informed consent.
2. Diagnosis of CD19+ B-cell lymphoma or leukemia (ALL or CLL).

* Subjects who previously received CD19-directed therapy must have biopsy or flow cytometry-confirmed CD19+ tumor following the CD19-directed therapy; may be performed by local laboratory.
3. The disease is:

Cohort A (non-ALL subjects):
* Relapsed or refractory after ≥2 lines of therapy, including a CD20 antibody, if an indolent lymphoma.
* Relapsed or refractory after ≥2 lines of therapy, including ibrutinib and venetoclax, if CLL.
* Relapsed or refractory after ≥2 lines of therapy, including a CD20 antibody and an anthracycline, and the subject is ineligible for autologous stem cell transplantation, if an aggressive or highly aggressive lymphoma.
* Ineligibility for autologous stem cell transplantation includes non-responsive disease after salvage therapy and failure to mobilize stem cells for transplant.

Cohort B (ALL subjects):
* Relapsed or refractory after ≥2 lines of therapy.
4. Measurable disease by current criteria (Lugano criteria for lymphomas, IWG criteria for CLL, and detectable disease for ALL).
5. Age 3 to 75 years; subjects \<18 years old will not be enrolled as the first subject on any dose level.
6. BSA ≤2.4 m2.
7. Bilirubin \<2 times the upper limit of normal (ULN) (3 times if the subject has Gilbert syndrome).
8. Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \<5 times ULN.
9. Estimated glomerular filtration rate (GFR) ≥50 mL/min.
10. Pulse oximetry ≥90% on room air.
11. Karnofsky or Lansky score ≥70.
12. Recovered from the acute toxic effects of all prior chemotherapy based on the enrolling physician's assessment (if some effects of chemotherapy are expected to last long term, for example, residual neuropathy, anemia, or alopecia, subject is eligible if meets other eligibility criteria).
13. Life expectancy \>12 weeks.

Exclusion Criteria:

Subjects who meet any of the following criteria will be excluded from this study:

1. Females who are breastfeeding or pregnant at Screening (as documented by a positive urine or serum pregnancy test; may be performed by local laboratory).
2. Females of childbearing potential who:

* Had unprotected sexual intercourse within 30 days before study entry and who do not agree to use a highly effective method of contraception (eg, total abstinence, an intrauterine device, a double-barrier method \[such as condom plus diaphragm with spermicide\], a contraceptive implant, an oral contraceptive, or have a vasectomized partner with confirmed azoospermia) through the Day 365 visit or for 30 days after study drug discontinuation.
* Are currently abstinent, and do not agree to use a double-barrier method (as described above) or refrain from sexual activity through the Day 365 visit or for 30 days after study drug discontinuation.
* Are using hormonal contraceptives but are not on a stable dose of the same hormonal contraceptive product for at least 4 weeks before dosing and who do not agree to use the same contraceptive through the Day 365 visit or for 30 days after study drug discontinuation.
3. Males who have not had a successful vasectomy (confirmed azoospermia) or they and their female partners do not meet the criteria above (ie, not of childbearing potential or practicing highly effective contraception through the Day 365 visit or for 30 days after study drug discontinuation). No sperm donation is allowed through the Day 365 visit or for 30 days after study drug discontinuation.
4. Currently receiving any investigational agents or received any cellular therapies within the previous 6 weeks prior to the KUR-502 infusion.
5. History of Grade 2 to 4 GvHD.
6. History of hypersensitivity reactions to murine protein-containing products.
7. Active infection with human immunodeficiency virus (HIV) or human T cell lymphotropic virus (HTLV).
8. Active infection with hepatitis B virus (HBV) or hepatitis C virus (HCV).
9. Uncontrolled active bacterial, fungal, or other viral infection.