For some "structures" (in informal sense for a lack of a formal term) in mathematics, such as groups, rings, and vector spaces, a bijective homomorphism is an isomorphism; i.e. the inverse is also a homomorphism. For some other structures, such as topological spaces and differentiable manifolds, a bijective homomorphism may not be an isomorphism.
Are there characterizations of sub-classes of structures which have the property that a bijective homomorphism is an isomorphism? For example, do all algebraic structures (in formal sense this time) have this property?