This is a Phase 1, non-randomized, dose-escalating, open label, multi-center study to be conducted in two parts (Part A and Part B). RO6870810 is a small molecule, non-covalent inhibitor of bromodomain and extra-terminal (BET) family of bromodomains. This study is designed to characterize the safety, tolerability, pharmacokinetics and anti-tumor activity of RO6870810 in participants with histologically confirmed solid tumors with progressive disease (PD) which is refractory or intolerant to standard/approved therapies. In Part A, RO6870810 will be administered by subcutaneous (SC) injection daily for either 21 consecutive days in a 28-day cycle or for 14 consecutive days in a 21-day treatment cycle in participants with advanced solid tumor malignancies to determine the maximum tolerated dose (MTD) and dose-limiting toxicities (DLT). In Part B, RO6870810 will be administered at a dose up to the MTD to further characterize the safety profile and biological effect in a subset of participants with advanced solid tumor malignancies. It is anticipated that a total of 84 participants will be enrolled in to this study (54 in Part A and 30 in Part B). In addition, it is expected that up to 20 participants with histologically confirmed nuclear protein in testis (NUT)-midline carcinoma (NMC) with progressive disease requiring therapy will be enrolled in the sub-study of Parts A and B. In addition, up to 20 participants with diffuse large B-cell lymphoma (DLBCL) may be enrolled at selected study sites.