Inclusion Criteria:

          1. Able to understand and comply with the protocol and study procedures, understand the
             risks involved in the study, and provide written informed consent before any
             study-specific procedure is performed.

          2. Men and women 18 years of age or older.

          3. Subjects with histologically or cytologically confirmed advanced solid tumors or
             lymphoma that is metastatic or unresectable, and for whom standard life-prolonging
             measures are not available. Specific tumor types that will be selected for study in
             Phase 2 are detailed in the protocol.

             a. For Phase 2 Cohort 3, subjects must have histologically confirmed PTCL (local
             pathology report) as defined by 2016 World Health Organization (WHO) classification.
             The following subtypes are eligible for the study: adult T-cell lymphoma/leukemia,
             extranodal natural killer (NK)/T-cell lymphoma nasal type, enteropathy-associated
             T-cell lymphoma, monomorphic epitheliotropic intestinal T-cell lymphoma, hepatosplenic
             T-cell lymphoma, subcutaneous panniculitis-like T-cell lymphoma, peripheral T-cell
             lymphoma not otherwise specified, angioimmunoblastic T-cell lymphoma, follicular
             T-cell lymphoma, nodal peripheral T-cell with T-follicular helper (THF) phenotype, and
             anaplastic large-cell lymphoma.

          4. For Phase 2 Cohorts 3 and 4, patients must have evidence of documented progressive
             disease and must have received at least two prior systemic therapies.

               1. Subjects with CD30-positive lymphoma must have received, be ineligible for, or
                  intolerant to brentuximab vedotin, provided that brentuximab vedotin is locally
                  approved and available.

               2. Subjects with mycosis fungoides or Sezary syndrome must have received, be
                  ineligible or intolerant to mogamulizumab, provided that mogamulizumab is locally
                  approved and available.

          5. In the Phase 2 portion of the protocol only, subjects must have measurable disease
             according to response criteria appropriate for their type of cancer.

             a. For Phase 2 Cohort 3 (PTCL), measurable disease by contrast-enhanced diagnostic CT
             (at least 1 nodal lesion >1.5 cm or extranodal lesions >1.0 cm) is required.

          6. Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2.

          7. Acceptable organ function, as evidenced by the following laboratory data:

               1. Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) <=2.0 * upper
                  limit of normal (ULN).

               2. Total serum bilirubin <=1.5 * ULN

               3. Absolute neutrophil count (ANC):

                    -  Phase 1 and 2 (except Phase 2 subjects with known lymphoma; ie, not
                       applicable for Cohorts 3 or 4) >=1500 cells/mm3

                    -  Phase 2 subjects with known lymphoma: >=1000 cells/mm3 (>750 cell/mm3 for
                       subjects with lymphoma in bone marrow)

               4. Platelet count:

                    -  Phase 1 and 2 (except Phase 2 subject with known lymphoma; ie, not
                       applicable for Cohorts 3 or 4) >=100,000 cells/mm3

                    -  Phase 2 subjects with known lymphoma: >= 50,000 cells/mm3; >=25,000
                       cells/mm3 for subjects with lymphoma in bone marrow

               5. Serum creatinine levels <= 1.5 * ULN, or calculated (by Cockcroft-Gault formula
                  or other accepted formula) or measure creatinine clearance >=50 mL/min.

               6. Amylase and lipase <=ULN.

          8. Women of child-bearing potential (according to recommendations of the Clinical Trial
             Facilitation Group [CTFG]; see protocol for details) must not be pregnant or
             breastfeeding and must have a negative pregnancy test at screening. Women of
             child-bearing potential and men with female partners of child-bearing potential must
             agree to practice 2 highly effective contraceptive measures of birth control (as
             described in the protocol) and must agree not to become pregnant or father a child
             while receiving treatment with study drug and for at least 3 months after completing
             treatment. Contraceptive measures which may be considered highly effective comprise
             combined hormonal contraception (oral, vaginal, or transdermal) or progestogen-only
             hormonal contraception (oral, injectable, implantable) associated with inhibition of
             ovulation, intrauterine device, intrauterine hormone-releasing system, bilateral tubal
             occlusion, sexual abstinence, and surgically successful vasectomy. Abstinence is
             acceptable only if it is consistent with the preferred and usual lifestyle of the
             subject. Periodic abstinence (eg, calendar, ovulation, symptothermal, or postovulation
             methods) and withdrawal are not acceptable methods of birth control.

        Exclusion Criteria:

          1. Hypersensitivity to ASTX660, excipients of the drug product, or other components of
             the study treatment regimen.

          2. Poor medical risk because of systemic diseases (e.g. active uncontrolled infections)
             in addition to the qualifying disease under study.

          3. Life-threatening illness, significant organ system dysfunction, or other condition
             that, in the investigator's opinion, could compromise subject safety or the integrity
             of the study outcomes, or interfere with the absorption or metabolism of ASTX660.

          4. History of, or at risk for, cardiac disease, as evidenced by 1 or more of the
             following conditions:

               1. Abnormal left ventricular ejection fraction (LVEF; <50%) or echocardiogram ECHO
                  or multiple gated acquisition scan (MUGA).

               2. Congestive cardiac failure of >= Grade 3 severity according to New York Heart
                  Association (NYHA) functional classification defined as subjects with marked
                  limitation of activity and who are comfortable only at rest.

               3. Unstable cardiac disease including angina or hypertension as defined by the need
                  for overnight hospital admission within the last 3 months (90 days).

               4. History or presence of complete left bundle branch block, heart block, cardiac
                  pacemaker or significant arrhythmia.

               5. Concurrent treatment with any medical that prolongs QT interval and may induce
                  torsades de pointes, and which cannot be discontinued at least 2 weeks before
                  treatment with ASTX660. [Applies to Phase 1 only].

               6. Personal history of long QTc syndrome or ventricular arrhythmias including
                  ventricular bigeminy.

               7. Screening 12-lead ECG with measurable QTc interval (according to either
                  Fridericia's or Bazett's correction) of >=470 msec).

               8. Any other condition that, in the opinion of the investigator, could put the
                  subject at increased cardiac risk.

          5. Known history of human immunodeficiency virus (HIV) infection, or seropositive results
             consistent with active hepatitis B virus (HBV) or active hepatitis C virus (HCV)
             infection.

          6. Grade 2 or greater neuropathy [Applies to Phase 1]. Grade 3 or greater neuropathy
             [Applies to Phase 2].

          7. Known brain metastases, unless stable or previously treated.

          8. Known significant mental illness or other conditions such as active alcohol or other
             substance abuse that, in the opinion of the investigator, predisposes the subject to
             high risk of noncompliance with the protocol treatment or assessments.

          9. Prior anticancer treatments or therapies within the indicated time window prior to
             first dose of study treatment (ASTX660), as follows:

               1. Cytotoxic chemotherapy or radiotherapy within 3 weeks prior and any encountered
                  treatment-related toxicities (excepting alopecia) not resolved to Grade 1 or less
                  [Phase 1] or Grade 2 or less [Phase 2].

               2. Skin directed treatments, including topicals and radiation within 2 weeks prior.

               3. Monoclonal antibodies within 4 weeks prior and any encountered treatment-related
                  toxicities not resolved to Grade 1 or less [Phase 1] or Grade 2 or less [Phase
                  2].

               4. Small molecules or biologics (investigational or approved) within the longer of 2
                  weeks or 5 half-lives prior to study treatment and any encountered
                  treatment-related toxicities not resolved to Grade 1 or less [Phase 1] or Grade 2
                  or less [Phase 2].

               5. At least 6 weeks must have elapsed since CAR-T infusion and subjects must have
                  experienced disease progression, and not have residual circulating CAR-T cells in
                  peripheral blood (based on local assessment). Any encountered treatment-related
                  toxicities must have resolved to Grade ≤1.

         10. Concurrent second malignancy currently requiring active therapy, except breast or
             prostate cancer stable on or responding to endocrine therapy or superficial bladder
             cancer [Phase 2].

         11. Known central nervous system (CNS) lymphoma [Phase 2].

         12. Patients with a history of allogenic transplant must not have ≥Grade 3
             graft-versus-host disease (GVHD) or any clinically significant GVHD requiring systemic
             immunosuppression [Phase 2].

         13. Systemic corticosteroids >20 mg prednisone equivalent (unless patient has been taking
             a continuous dose for >3 weeks prior to study entry and there is documented
             radiological progression) [Phase 2]. Stable dose of medium or low potency topical
             corticosteroids for at least 3 weeks prior to study entry are permitted [Phase 2].