Inclusion Criteria:
* Must be ≥18 years of age at the time the informed consent form (ICF) is signed.
* Has a histologically or cytologically confirmed adenocarcinoma of the prostate.
* Has either of the following:
1. Measurable disease per RECIST v1.1 (with PCWG3 modifications)
2. Evaluable disease per PCWG4, as assessed by the local site Investigator/radiology
3. For Part 2.1B (omPC expansion) only: PSMA-avid disease on PSMA-PET, defined as lesion(s) demonstrating uptake above blood pool (PSMA score ≥1) and assessed by the Investigator as consistent with metastatic prostate cancer, in the setting of biochemical recurrence (PSA ≥0.2 ng/mL).
Exclusion Criteria:
* Has a diagnosis of immunodeficiency.
* Has had a prior stem cell, bone marrow, or organ transplant.
* Has a known history of human immunodeficiency virus (HIV) infection.
* Has active or chronic hepatitis B virus (HBV), or hepatitis C virus (HCV) infection.
* Has an active autoimmune disease (non-immunotherapy induced conditions) that has required systemic treatment in the past 2 years (ie, with use of disease modifying agents, corticosteroids, or immunosuppressive drugs).
* Has a history of (noninfectious) pneumonitis that required steroids or current active organizing pneumonitis/interstitial lung disease.
* Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Individuals with previously treated CNS metastases may participate in JUR-003 monotherapy cohorts (Parts 1.1, 2.1, and 1.3 where applicable) provided they are radiologically stable (ie, without evidence of progression for at least 2 weeks by repeat imaging \[note that the repeat imaging should be performed during study Screening\]), clinically stable, and without requirement of steroid treatment for at least 14 days prior to first dose of study drug.
* Has untreated spinal cord compression. Individuals must be neurologically stable off steroids for at least 4 weeks prior to first dose of study drug.
* Has a history of a previous secondary malignancy within 3 years of Screening (except basal cell or squamous cell carcinoma of the skin or carcinoma in situ treated with curative therapy or other localized, low-grade tumors deemed cured, or whose natural history does not have the potential to interfere with the safety or efficacy assessments of the current study and not treated with systemic anticancer therapy \[except hormonal therapy\]).
* Has a known psychiatric or substance abuse disorder that would interfere with the individuals' ability to cooperate with the requirements of the study.
* Has a history of venous thromboembolic events (eg, pulmonary embolism) within 1 month prior to the first dose of JUR-003. Uncomplicated (Grade ≤2) deep vein thrombosis is not considered exclusionary.
