Inclusion Criteria:

          -  Radiologically measurable and clinically evaluable disease

          -  Absence of rapid disease progression or threat to vital organs or critical anatomical
             sites requiring urgent alternative medical intervention

          -  Confirmed at least one tumor lesion with location accessible to safely biopsy per
             clinical judgment (special requirements apply for Part C; Participants with only one
             target lesion and no non-target lesions can enroll after documented agreement with the
             Medical Monitor).

          -  Life expectancy of greater than or equal to (>=12) weeks

          -  Eastern Cooperative Oncology Group (ECOG) Performance Status 0-1

          -  Participants with unilateral pleural effusion (other than non-small cell lung cancer
             [NSCLC] indication) should fulfill the following criteria for pulmonary and cardiac
             functions: Global Initiative for Chronic Obstructive Lung Disease (GOLD)
             classification 0 - 1 level and New York Heart Association (NYHA) classification class
             1 or better

          -  Forced expiratory volume 1 (FEV1) >70% and forced vital capacity (FVC) >70% of
             predicted value; participants with lung metastases should present with DLCO >60% of
             predicted value

          -  Adequate cardiovascular, hematological, liver and renal function

          -  All acute toxic effects of any prior radiotherapy, chemotherapy, or surgical procedure
             must have resolved to grade less than or equal to (<=) 1, except alopecia (any grade)
             and Grade 2 peripheral neuropathy

          -  Negative serum pregnancy test within 7 days prior to study treatment in premenopausal
             women and women less than (<) 12 months after menopause

          -  For women who are not postmenopausal and have not undergone surgical sterilization:
             agreement to remain abstinent or use two adequate non-hormonal methods of
             contraception, including at least one method with a failure rate of <1 percent (%) per
             year, during the treatment period and for a period of time after the last dose of
             study drug(s) as defined in the protocol

          -  For men: agreement to remain abstinent or use contraceptive measures and agreement to
             refrain from donating sperm during the treatment period and for at least for at least
             2 months after the last dose of study treatment

          -  For Part A exclusively (RO6874281 monotherapy), confirmed advanced and/or metastatic
             solid tumor, with at least one tumor lesion of location accessible to biopsy per
             clinical judgment of the treating physician, and confirmed progression at baseline;
             for whom no standard therapy that would confer clinical benefit to the participant
             exists

          -  For Part B exclusively (RO6874281 in combination with trastuzumab), participants with
             metastatic or recurrent or locally advanced human epidermal growth factor receptor 2
             (HER2)-positive breast cancer, as defined by the College of American Pathologists HER2
             testing guidelines, who have progressed on at least two lines of HER2-directed
             therapies in the metastatic setting and the last therapy prior to going on study has
             to contain a HER2-directed antibody; baseline left ventricular ejection fraction
             (LVEF) of >=50% (measured by echocardiography) predose on Cycle 1 Day 1

          -  For Part C exclusively (RO6874281 in combination with cetuximab), participants with
             recurrent, unresectable or metastatic squamous cell carcinoma of the head and neck.
             Participants can have had standard or experimental treatment, including but not
             limited to radiation therapy, chemotherapy, or immunotherapy

          -  Participants with Gilbert's syndrome will be eligible for the study

        Exclusion Criteria:

          -  History of, active, or suspicion of autoimmune disease (exceptions apply)

          -  Adverse events from prior anti-cancer therapy that have not resolved to Grade 1,
             except for alopecia, vitiligo, or endocrinopathies managed with replacement therapy

          -  Symptomatic or untreated central nervous system (CNS) metastases

          -  History of treated asymptomatic CNS metastases with any of the following: Metastases
             to the brain stem, midbrain, pons, medulla, cerebellum, or within 10 millimeters (mm)
             of the optic nerves and chiasm; history of intracranial or spinal cord hemorrhage;
             lacking radiographic demonstration of improvement upon the completion of CNS-directed
             therapy and evidence of progression between completion of therapy and the baseline
             radiographic study; ongoing requirement for dexamethasone; stereotactic or whole brain
             radiation within 28 days before the start of study treatment; last CNS radiographic
             study less than 4 weeks since completion of radiotherapy and less than 2 weeks since
             the discontinuation of corticosteroids; CNS metastases treated by resection or brain
             biopsy performed within 28 days before the start of study treatment

          -  Participants with an active second malignancy

          -  Evidence of significant, uncontrolled concomitant diseases that could affect
             compliance with the protocol or interpretation of results, including diabetes
             mellitus, history of relevant pulmonary disorders, and known autoimmune diseases or
             other disease with ongoing fibrosis

          -  Participants (all indications) with confirmed bilateral pleural effusion and NSCLC
             participants with confirmed uni- or bilateral pleural effusion by X-ray are not
             eligible

          -  Significant cardiovascular/cerebrovascular disease within 6 months prior to Day 1 of
             study drug administration

          -  Active or uncontrolled infections

          -  Known human immunodeficiency virus (HIV) or known active hepatitis B virus or
             hepatitis C virus infection

          -  History of chronic liver disease or evidence of hepatic cirrhosis

          -  Any other diseases, metabolic dysfunction, physical examination finding, or clinical
             laboratory finding that give reasonable suspicion of a disease or condition that would
             contraindicate the use of an investigational drug

          -  Major surgery or significant traumatic injury <28 days prior to the first RO6874281
             infusion (excluding biopsies) or anticipation of the need for major surgery during
             study treatment

          -  Dementia or altered mental status that would prohibit informed consent

          -  Pregnant or breastfeeding women

          -  Known hypersensitivity to any of the components of RO6874281

          -  Concurrent therapy with any other investigational drug

          -  Immune-related endocrinopathies

          -  Immunomodulating agents <28 days prior to first dose of study drug

          -  Treatment with systemic immunosuppressive medications

          -  Severe dyspnea at rest due to complications of advanced malignancy or requiring
             supplementary oxygen therapy

          -  For Part B exclusively, known hypersensitivity to any of the components of trastuzumab

          -  For Part C exclusively, known hypersensitivity to any of the components of cetuximab

          -  For Parts A, B, and C, eligibility of participants who require blood transfusion
             before and after the start of the study treatment should be discussed by the Sponsor
             and investigator

          -  For Parts B and C, Participant eligibility for treatment with trastuzumab or cetuximab
             should be verified against trastuzumab or cetuximab labeling documents.