The impressive success of chimeric antigen receptor (CAR)-T cell therapies in treating advanced B-cell malignancies has spurred a frenzy of activity aimed at developing CAR-T therapies for other cancers, particularly solid tumors, and optimizing engineered T cells for maximum clinical benefit in many different disease contexts. A rapidly growing body of design work is examining every modular component of traditional single-chain CARs as well as expanding out into many new and innovative engineered immunoreceptor designs that depart from this template. New approaches to immune cell and receptor engineering are being reported with rapidly increasing frequency, and many recent high-quality reviews (including one in this special issue) provide comprehensive coverage of the history and current state of the art in CAR-T and related cellular immunotherapies. In this review, we step back to examine our current understanding of the structure-function relationships in natural and engineered lymphocyte-activating receptors, with an eye towards evaluating how well the current-generation CAR designs recapitulate the most desirable features of their natural counterparts. We identify key areas that we believe are under-studied and therefore represent opportunities to further improve our grasp of form and function in natural and engineered receptors and to rationally design better therapeutics.