For a partially ordered set P, we denote by Co (P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co (P) iff L satisfies (S) , (U) , and (B) . Furthermore, if L has an embedding into some Co (P) , then it has such an embedding that preserves the existing bounds. If L is finite, then one can take P finite, with |P|⩽2 J(L) 2−5 J(L) +4, where J( L) denotes the set of all join-irreducible elements of L. On the other hand, the partially ordered set P can be chosen in such a way that there are no infinite bounded chains in P and the undirected graph of the predecessor relation of P is a tree.