Clathrin is a molecular scaffold for vesicular uptake of cargo at the plasma membrane, where its assembly into cage-like lattices underlies the clathrin-coated pits of classical endocytosis. This review describes the structures of clathrin, major cargo adaptors, and other proteins that participate in forming a clathrin-coated pit, loading its contents, pinching off the membrane as a lattice-enclosed vesicle, and recycling the components. It integrates as much of the structural information as possible at the time of writing into a sketch of the principal steps in coated-pit and coated-vesicle formation.