A graph $G$ is a minimal claw-free graph (m.c.f. graph) if it contains no $K_{1,3}$ (claw) as an induced subgraph and if, for each edge $e$ of $G$, $G-e$ contains an induced claw. We investigate properties of m.c.f. graphs, establish sharp bounds on their orders and the degrees of their vertices, and characterize graphs which have m.c.f. line graphs.