This Study Aims to Find a Safe and Effective Dose of BI 754091. The Study Also Aims to Find Safe and Effective Doses of BI 754091 and BI 754111 in Combination. This Study is Done in Asian Patients With Different Types of Cancer

Inclusion Criteria: * Of full age (according to local legislation) at the time of signing of the informed consent form (ICF) * Women of childbearing potential (WOCBP)1 with negative serum pregnancy test at screening and men able to father a child, who agree 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. A list of contraception methods meeting these criteria is provided in the patient information. The requirement of contraception does not apply to women of no childbearing potential but they must have an evidence of such at screening * Signed and dated written informed consent in accordance with ICH-GCP and local legislation prior to admission to the trial * Patients with measurable lesions according to RECIST v1.1 * Conditions specific to respective part of the trial: * Part I (BI 754091 dose-finding part): * Patients with a confirmed diagnosis of advanced, unresectable, and/or metastatic solid tumours (any type) * For whom no therapy of proven efficacy exists, or who are not amenable to standard therapies. * Previous treatment with an anti-PD-1 mAb is allowed as long as the last administration of the anti PD-1 mAb on the previous treatment is a minimum of 28 days prior to the first BI 754091 treatment. * Part II (Combination dose-finding part): * Patients with a confirmed diagnosis of advanced, unresectable, and/or metastatic solid tumours (any type) * For whom no therapy of proven efficacy exists, or who are not amenable to standard therapies. * Previous treatment with an anti-PD-1 mAb is allowed as long as the last administration of the anti PD-1 mAb on the previous treatment is a minimum of 28 days prior to the first BI 754091 treatment. * Part III (Expansion part): * Cohort A: Patients with gastric/esophagogastric junction cancer, with no prior treatment with anti-PD-1/PD-L1 antibody, and who received at least one line of systemic medical treatment excluding adjuvant therapy * Cohort B: Patients with esophageal cancer with no prior treatment with anti-PD-1/PD-L1 antibody, and who received at least one line of systemic medical treatment excluding adjuvant therapy * Cohort C: Patients with hepatocellular cancer with no prior treatment with anti-PD-1/PD-L1 antibody, who received at least one line of systemic medical treatment excluding adjuvant therapy, and whose Child-Pugh score is 7 or less * Cohort D: Patients with gastric/esophagogastric junction cancer, esophageal cancer, or hepatocellular cancer with a prior treatment with anti-PD-1/PD-L1 antibody * Cohort E: First line squamous or non-squamous NSCLC patients: * Without EGFR mutations or ALK rearrangements * PD-L1 expression level \<50% * All cohorts: Patients with advanced and/or metastatic disease, with at least 1 tumour lesion amenable to biopsy, and must be medically fit for biopsy at screening as determined by investigator and willing to undergo a biopsy before first treatment (if adequate archival tissue is not available) and, unless clinically contraindicated, after 6 weeks on therapy. * Eastern Cooperative Oncology Group (ECOG) performance status (PS): 0 to 1 at screening Exclusion Criteria: * Major surgery (major according to the investigator's assessment) performed within 12 weeks prior to study entry or planned within 12 months after screening, e.g. hip replacement * Patients who must or wish to continue the intake of restricted medications or any drug considered likely to interfere with the safe conduct of the trial * Previous treatment with study medications in this trial * Any investigational or anti-tumour treatment within 4 weeks or 5 half-life periods (whichever is shorter) prior to the initiation of trial treatment * Any unresolved toxicities from prior therapy greater than Common Terminology Criteria for Adverse Events (CTCAE) Grade 1 at the time of starting study treatment with the exception of alopecia and Grade 2 neuropathy due to prior platinum-based therapy * (Part II and III only) Prior treatment with anti-LAG-3 agents * Patients with lung cancer that have epidermal growth factor receptor (EGFR) mutations or anaplastic lymphoma kinase (ALK) rearrangements, unless disease has progressed following available EGFR or ALK targeted therapy * Presence of other active invasive cancers other than the one treated in this Trial within 5 years Prior to screening, with the exception of appropriately treated basal or squamous-cell carcinoma of the skin or in situ carcinoma of the uterine cervix, or other local tumours considered cured by local treatment * Untreated brain metastasis(es) that may be considered active. Patients with previously treated brain metastases may participate provided they are stable (i.e., without evidence of PD by imaging for at least 4 weeks prior to the first dose of trial treatment, and any neurologic symptoms have returned to baseline), and there is no evidence of new or enlarging brain metastases * Inadequate organ function or bone marrow reserve as demonstrated by the following laboratory values: * Absolute neutrophil count \<1.5 x 10\^9/L (\<1500/mm\^3) * Platelet count \<100 x 10\^9/L (\<100,000/mm\^3) * Haemoglobin \<9.0 g/dL * Alanine aminotransferase (ALT) \>2.5 times the upper limit of normal (ULN) if no demonstrable liver lesion(s) (primary or metastases) or \>5 times ULN in the presence of liver lesion(s) * Aspartate aminotransferase (AST) \>2.5 times ULN if no demonstrable liver lesion(s) or \>5 times ULN in the presence of liver lesion(s) * Total bilirubin \>1.5 times ULN, except for patients with Gilbert's syndrome who are excluded if total bilirubin \>3.0 times ULN or direct bilirubin \>1.5 times ULN * Serum creatinine (measured by enzymatic assay, Isotope dilution mass spectroscopy \[IDMS\] standardized Jaffe assay, or non-IDMS Jaffe assay) \>1.5 times ULN or estimated glomerular filtration rate (eGFR) \<30mL/min/1.73m\^2 (Chronic Kidney Disease Epidemiology \[CKD-EPI\] Collaboration equation); confirmation of eGFR is only required when creatinine is \>1.5 X ULN * International normalized ratio (INR) (only tested if clinically indicated) \>1.5 times ULN (if treated with anticoagulants, prolonged INR is acceptable) * Any of the following cardiac criteria: * Mean resting corrected QT interval (QTc) \>470 msec * Any clinically important abnormalities (as assessed by the investigator) in rhythm, conduction, or morphology of resting ECGs, e.g., complete left bundle branch block, third degree heart block * Any factors that increase the risk of QTc prolongation or risk of arrhythmic events such as heart failure, hypokalaemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years-of-age, or any concomitant medication known to prolong the QT interval * Patients with an ejection fraction \<55% or the lower limit of normal of the institutional standard will be excluded. Only in cases where the Investigator (or the treating physician or both) suspects cardiac disease with negative effect on the EF, will the EF be measured during screening using an appropriate method according to local standards to confirm eligibility (e.g., echocardiogram \[ECHO\], multi-gated acquisition scan \[MUGA\]). A historic measurement of EF no older than 6 months prior to first administration of study drug can be accepted provided that there is clinical evidence that the EF value has not worsened since this measurement in the opinion of the Investigator or of the treating physician or both. * History of pneumonitis within the last 5 years * History of severe hypersensitivity reactions to other mAbs * History of severe hypersensitivity reactions to the ingredients of study drug * Immunosuppressive corticosteroid doses (\>10 mg prednisone daily or equivalent) within 4 weeks prior to the first dose of study treatment * Active autoimmune disease or a documented history of autoimmune disease, except vitiligo or resolved childhood asthma/atopy, or asthma well controlled with steroids * Active infection requiring systemic treatment (antibacterial, antiviral, or antifungal therapy) at start of treatment in this trial * Known history of human immunodeficiency virus (HIV) infection or laboratory evidence of hepatitis virus infection with positive results of hepatitis B surface (HBs) antigen and/or presence of HBc antibody together with HBV-DNA and/or hepatitis C RNA (HIV and hepatitis test results obtained in routine diagnostics are acceptable if done within 14 days before the informed consent date). However, for patients with hepatocellular cancer in Part III Cohorts C and D, patients with HBV and/or HCV infection are allowed. Hepatocellular cancer patients in Part III Cohorts C and D with HBV infection must be receiving effective antiviral therapy (viral load \<100 IU/mL) * Current or history of interstitial lung disease * Chronic alcohol or drug abuse or any condition that, in the investigator's opinion, makes him/her an unreliable trial patient, unlikely to complete the trial, or unable to comply with the protocol procedures. However, for patients with hepatocellular cancer in Part III Cohorts C and D, past chronic alcohol abuse are allowed * Women who are pregnant, nursing, or who plan to become pregnant during the trial