Inclusion Criteria:  Phase 1 Dose Escalation only:  - Participant has relapsed/refractory (R/R) solid tumor known to express NY-ESO-1 after completing available Standard of Care (SOC) therapy or is not a candidate for SOC therapy. NY-ESO-1 expression status is not required for participant entry.  Phase 2 Single agent and Combination Therapy only:  Participant has relapsed/refractory (R/R) synovial sarcoma (SS) or myxoid/round cell liposarcoma (MRCL) disease after undergoing available SOC treatment or is not a candidate for SOC therapy (must have previously received either an anthracycline or ifosfamide containing regimen or another systemic regimen, if not a candidate for either agent).  Participant has not received prior checkpoint inhibitor therapy (i.e., Programmed Cell Death Protein 1 [PD-1]/Programmed Death Ligand 1 [PD-L1] treatment naive) SS: confirmation by the presence of a translocation between SYT on the X chromosome and SSX1, SSX2, or SSX4 on chromosome 18 (may be presented in the pathology report as t [X;18]). MRCL: confirmation by the presence of the reciprocal chromosomal translocation t (12;16) (q13;p11) or t(12;22)(q13;q12).  Participant has R/R ovarian cancer that is:  platinum resistant OR platinum-sensitive, but the participant is not a candidate for platinum or other SOC therapy. Participant has not received prior checkpoint inhibitor therapy (i.e., naive PD-1/PD-L1 treatment participants). Participant has R/R solid tumor (melanoma, non-small cell lung cancer [NSCLC]-adenocarcinoma and squamous cell, or esophageal squamous cell carcinoma [ESCC]) after available SOC treatment or is not a candidate for SOC therapy (single-agent only). Participant consents to provide an archival tumor specimen in a tissue block or unstained serial slides prior to IP administration. Participant in phase 2 consents to provide tumor specimen obtained within 56 days prior to first dose of study treatment, as tissue block or unstained serial slides. Participant in phase 2 consents to undergoing a tumor biopsy (core needle biopsy or excision) during the treatment period. Participant has an Eastern Cooperative Oncology Group (ECOG) performance status of <= 2. Participant with life expectancy of >= 12 weeks at the time of screening. Participant must meet criteria for clinical laboratory tests during screening period.  A female participant is eligible to participate if she is not pregnant and at least one of the following conditions apply:  Not a woman of childbearing potential (WOCBP) OR WOCBP who agrees to follow the contraceptive guidance throughout the treatment period and for at least 6 months after the final investigational product (IP) administration. Female participant must not be breastfeeding at screening or during the study period and for 6 months after the final IP administration. Female participant must not donate ova at screening and throughout the study period and for 6 months after the final IP administration. A male participant with female partner(s) of childbearing potential must agree to use contraception during the treatment period and for at least 6 months after the final IP administration. Male participant must not donate sperm starting at screening and throughout the study period and for 6 months after the final IP administration. Participant agrees not to participate in another interventional study while on treatment. Participant has at least 1 measurable lesion per immune response evaluation criteria in solid tumors (iRECIST). The measurable lesion must be outside the field of radiation if participant had prior radiotherapy < 3 months from the completion of radiation.  Exclusion Criteria:  Participant has persistent non-hematological toxicities of >= grade 2 (National Cancer Institute's Common Terminology Criteria for Adverse Events [NCI-CTCAE] version 5.0), with symptoms and objective findings from treatment (including chemotherapy, kinase inhibitors, immunotherapy, experimental agents, radiation or surgery).  Participant has received any of the following therapies (for inclusion in the study, all abnormalities must have returned to <= grade 1):  Systemic immunomodulators (checkpoint inhibitors)-except the NY-ESO-1 solid tumor (melanoma, NSCLC-adenocarcinoma and squamous cell and ESCC) cohorts, which may have received prior checkpoint inhibitor therapy Immunosuppressive drugs including steroids <= 14 days prior to treatment Cytotoxic agents <= 14 days prior to treatment Investigational agent <= 21 days prior to treatment or 5 half-lives, whichever is shorter Radiation therapy <= 21 days prior to treatment Participant has clinically active or untreated nervous system metastases. Participants with previously treated Central Nervous System (CNS) metastases are eligible, if they are clinically stable and have no evidence of CNS progression by imaging for at least 4 weeks prior to start of study treatment and are not requiring immunosuppressive doses of systemic steroids (> 30 mg per day of hydrocortisone or > 10 mg per day of prednisone or equivalent) for longer than 2 weeks. Participant has an active autoimmune disease. Participant with type 1 diabetes mellitus, endocrinopathies stably maintained on appropriate replacement therapy, or skin disorders (e.g., vitiligo, psoriasis, or alopecia) not requiring systemic treatment are allowed. Participant was discontinued from prior immunomodulatory therapy due to a grade >= 3 toxicity that was mechanistically related (e.g., immune related) to the agent. Participant has known history of serious hypersensitivity reaction to a known ingredient of ASP0739 or pembrolizumab or severe hypersensitivity reaction to treatment with another monoclonal antibody. Participant has a prior malignancy active (i.e., requiring treatment or intervention) within the previous 2 years prior to screening visit, except for locally curable malignancies that have been apparently cured, such as basal or squamous cell skin cancer, superficial bladder cancer or carcinoma in situ of the cervix or breast. Participant has received a prior allogeneic bone marrow or solid organ transplant. Participant has an active uncontrolled infection within 14 days of treatment. Participant is known to have human immunodeficiency virus infection. Participant has active hepatitis B or C or other active hepatic disorder or participant is on hepatitis treatment. Hepatitis C RNA testing is not required in participants with negative hepatitis C antibody testing. Participant has any condition which makes the participant unsuitable for study participation. Participant has had a major surgical procedure and has not completely recovered within 28 days prior to the start of study treatment. Participant has had a myocardial infarction or unstable angina within 6 months prior to the start of study treatment or currently has an uncontrolled illness including, but not limited to, symptomatic congestive heart failure, clinically significant cardiac disease, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. Participant is expected to require another form of anti-cancer therapy while on study treatment. Participants for the combination therapy arm cohorts must not have known microsatellite instability or deficient MisMatch Repair.  Additional Exclusion Criteria for Participants in Combination Therapy Cohorts  Participants with a history of myocarditis or congestive heart failure (as defined by New York Heart Associated Functional Classification III or IV), as well as unstable angina, serious uncontrolled cardiac arrhythmia, uncontrolled infection, or myocardial infarction 6 months prior to study entry. Participants with active interstitial lung disease (ILD)/pneumonitis or a history of ILD/pneumonitis requiring treatment with systemic steroids. Participants with baseline pulse oximetry < 92% "on Room air."