T cells engineered to express chimeric antigen receptors (CAR-T cells) have shown impressive clinical efficacy in the treatment of B cell malignancies. However, the development of CAR-T cell therapies for solid tumors is hampered by the lack of truly tumor-specific antigens and poor control over T cell activity. Here we present an avidity-controlled CAR (AvidCAR) platform with inducible and logic control functions. The key is the combination of (i) an improved CAR design which enables controlled CAR dimerization and (ii) a significant reduction of antigen-binding affinities to introduce dependence on bivalent interaction, i.e. avidity. The potential and versatility of the AvidCAR platform is exemplified by designing ON-switch CARs, which can be regulated with a clinically applied drug, and AND-gate CARs specifically recognizing combinations of two antigens. Thus, we expect that AvidCARs will be a highly valuable platform for the development of controllable CAR therapies with improved tumor specificity.