This open-label, multi-center, dose-escalation study will evaluate the safety, pharmacokinetics, and therapeutic activity of RO6895882 in participants with Carcinoembryonic Antigen (CEA)-positive solid tumors who have progressed on the standard of care therapy. The study will be conducted in 3 parts. Part 1 will be a single ascending dose study in single participant cohort at low RO6895882 dose (less than or equal to [</=] 6 milligrams [mg]). Part 2 will be a dose-escalation study of RO6895882 monotherapy given every week (qw), every 2 weeks (q2w), and possibly every 3 weeks (q3w). Part 3 will be an expansion phase of the qw, q2w, and possibly q3w at maximum tolerated dose (MTD) (as determined in Part 2). Part 3 will only be conducted if the risk/benefit assessment, as evaluated by the Sponsor and the investigators, is in favor of the participants. Participants will be treated until disease progression, unacceptable toxicity or withdrawal from treatment for other reasons or death for a maximum duration of 24 months.