A Study of Zasocitinib in Adults With Psoriatic Arthritis Who Have or Have Not Been Treated With Biologic Medicines

Psoriatic arthritis (PsA) is a chronic inflammatory disease that affects the joints and skin in people who have psoriasis (PsO). The main aim of the study is to know how well zasocitinib (TAK-279) works in participants with active PsA based on their previous experience with specific treatments. The participants will be treated with either zasocitinib, or placebo. Participants will be in the study for up to 60 weeks.

Inclusion Criteria: Age: 1. The participant is aged 18 years or older at the time of signing the informed consent form (ICF). Disease Characteristics: 2. The participant has a diagnosis of PsA. 3. The participant must have signs and symptoms of PsA for at least 3 months prior to screening. 4. The participant meets the Classification Criteria for Psoriatic Arthritis (CASPAR criteria). 5. The participant has active arthritis as shown by a minimum of \>=3 tender joints in TJC68 and \>=3 swollen joints in SJC66 at the screening and baseline (Day 1) visits. 6. The participant has at least 1 active lesion of plaque PsO \>=2 cm in diameter, or any nail or nail bed changes characteristic of PsO. Medications for PsA: 7. The participant has had at least one of the following: 1. Inadequate response to a nonsteroidal anti-inflammatory drug (NSAID) (not applicable in the European Union \[EU\]/ European Economic Area \[EEA\]), OR 2. Inadequate response to a conventional synthetic disease-modifying antirheumatic drug (csDMARD), OR 3. Biological disease-modifying antirheumatic drug (DMARD)-inadequate response (Bio-IR): Inadequate response to up to 2 biologic DMARDs. Exclusion Criteria: PsA and PsO: 1. The participant has other disease(s) that might confound the evaluations of benefit of zasocitinib therapy, including but not limited to rheumatoid arthritis, axial spondyloarthritis, systemic lupus erythematosus, Lyme disease, gout, or fibromyalgia. 2. The participant has a concomitant comorbid skin condition that, in the opinion of the investigator, would interfere with the study assessments, such as evidence of non-plaque PsO (erythrodermic, pustular, predominately guttate PsO, inverse, or drug-induced PsO).