This study is a first-in-human, open-label, multi-center, dose-escalation Phase 1 clinical study of single-agent RO6958688 in patients with locally advanced and/or metastatic CEA-positive solid tumors who have progressed on standard treatment, are intolerant to standard of care (SOC), and/or are non-amenable to SOC. The study will be conducted in two parts. Part 1 will investigate the safety and pharmacokinetics of a single dose of RO6958688 in single patient cohorts. Part 2 will establish the appropriate therapeutic dose(s) based on safety, pharmacokinetics, and the maximum tolerated dose of RO6958688 for the weekly schedule. The anticipated time on treatment is a maximum of 24 months. The overall target sample size is 90.