A Study to Test BI 765179 Alone and in Combination With Ezabenlimab in Patients With Advanced Cancer (Solid Tumors) and BI 765179 in Combination With Pembrolizumab in Patients With Advanced Head and Neck Cancer

Inclusion Criteria: All cohorts: * Patients with locally advanced, unresectable or metastatic solid tumors who are either refractory after standard therapy for the disease or for whom standard therapy is not appropriate * Tumor with expected high expression of Fibroblast activation protein (FAP) of the following histologies: * Non-small cell lung carcinoma (NSCLC) * Gastric cancer * Esophageal adenocarcinoma or squamous cell carcinoma * Urothelial bladder carcinoma * Head and neck squamous cell carcinoma * Cutaneous malignant melanoma * Cutaneous squamous cell carcinoma * Hepatocellular carcinoma * Pancreatic adenocarcinoma * Colorectal cancer * Malignant pleural mesothelioma * Cervical squamous cell cancer * Ovarian carcinoma * Triple-negative breast cancer * At least 18 years of age at the time of the consent or over the legal age of consent in countries where that is greater than 18 years * Signed and dated, written informed consent (IC) in accordance with ICH-GCP and local legislation prior to admission to the trial * At least one measurable lesion outside of central nervous system (CNS) as defined per modified Response evaluation criteria in solid tumors (RECIST) v1.1 * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Adequate liver, bone marrow and renal organ function * Male or female patients. Women of childbearing potential (WOCBP) and men able to father a child must be ready and able to use highly effective methods of birth control per ICH M3 (R2) that result in a low failure rate of less than 1% per year when used consistently and correctly. These methods must be used during the study and for at least 6 months after the last dose of the study medication. A list of contraception methods meeting these criteria is provided in the patient information. * Patients with brain metastases are eligible provided they meet all of the following criteria: * Brain metastases have adequately been treated and are considered stable by the Investigator * Radiotherapy or surgery for brain metastases was completed at least 2 weeks prior to the first administration of BI 765179 * Patient is off steroids for at least 7 days (physiologic doses of steroids is permitted, if this was stable for the last 4 weeks) * The patient is off anti-epileptic drugs for at least 7 days Back-fill cohorts only: * Patient has agreed to and signed an IC form to provide mandatory pre-treatment and on-treatment fresh tumor biopsy * At least one lesion (separate from the evaluable target lesion outside of the CNS as defined per RECIST v1.1) that is accessible for mandatory paired pre and on-treatment biopsy Phase 1b: * Histologically or cytologically confirmed diagnosis of metastatic or incurable, recurrent head and neck squamous cell carcinoma (HNSCC) * No prior systemic therapy administered in the metastatic or incurable, recurrent setting * Primary tumor locations of oropharynx, oral cavity, hypopharynx, or larynx * At least 18 years of age at the time of the consent or over the legal age of consent in countries where that is greater than 18 years * Signed and dated written IC in accordance with ICH-GCP and local legislation prior to admission to the trial Further inclusion criteria apply Exclusion Criteria: Phase 1a * Currently enrolled in another investigational device or drug trial * Previous or concomitant malignancies other than the one treated in this trial within the last 2 years except: * Effectively treated non-melanoma skin cancers * Effectively treated carcinoma in situ of the cervix * Effectively treated ductal carcinoma in situ * Other effectively treated malignancy that is considered cured by 'local treatment' * Previous treatment with agents targeting CD137 * Known leptomeningeal disease or spinal cord compression due to disease * Anticoagulant treatment that cannot be safely interrupted if medically needed (e.g., biopsy) based on the opinion of the Investigator * Persistent toxicity from previous treatments that has not resolved to ≤ Common terminology criteria for adverse events (CTCAE) Grade 1 (except for alopecia, CTCAE Grade 2 neuropathy, asthenia/fatigue or grade 2 endocrinopathies controlled by replacement therapy) * Patient has a diagnosis of immunodeficiency * Patient with history of immunosuppressive medication within 14 days prior to the first dose of BI 765179. The following are exceptions to this criterion: * Use of intranasal, inhaled, or topical corticosteroids, local steroid injections (e.g., intra-articular injections) * Systemic corticosteroids at physiologic doses ≤10 mg/day (prednisone or equivalent) * Physiological replacement dose of corticosteroids Further exclusion criteria apply. Phase Ib * Disease suitable for local therapy administered with curative intent * Participants must not have a primary tumor site of nasopharynx or sino-nasal cancer or salivary gland cancers (any histology) * Currently enrolled in another investigational device or drug trial * Life expectancy of \<3 months and/or has rapidly progressing disease * Diagnosed and/or treated additional malignancy within 2 years prior to randomization with the exception of curatively treated basal cell carcinoma of the skin, squamous cell carcinoma of the skin, and/or curatively resected in situ cervical and/or in situ breast cancers Further exclusion criteria apply