Convex geometries are defined by the authors as closure spaces satisfying the anti-exchange axiom. In the finite (resp. atomistic) case, the associated lattices of closed sets are join-semidistributive (with respect to atoms) and lower semimodular. The paper focuses on the representation of join-semidistributive lattices within appropriate convex geometries having a join-semidistributive lattice of closed sets, and on developing a geometric framework for the latter. The most remarkable results are the following. Each finite join-semidistributive lattice can be embedded into a convex geometry on the set of join irreducibles as well as into the lattice of algebraic subsets of a lattice which is both algebraic and dually algebraic. The quasivariety of join-semidistibutive lattices is generated by each of the following classes: its finite members; the lattices of closed subsets of finite atomistic convex geometries; the lattices of quasivarieties. In the third section, the authors study examples of convex geometries having a join-semidistributive lattice. The most important are: those associated with partial orders resp. graphs; the lattices of convex compact subsets of Euclidean space; the lattices of finitely generated meet-subsemilattices. Finally, the following representation theorems for infinite join-semidistributive lattices are obtained: each can be embedded into the lattice of a convex geometry which is atomistic, algebraic, and biatomic; in the finitely presented case there is a representation within an dually algebraic, dually spatial, and join-semidistributive lattice. The paper is completed with 5 problems which gave direction for much of the present research.