TITLE: When is a bijective homomorphism an isomorphism?
QUESTION [11 upvotes]: 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?

REPLY [4 votes]: A relevant anecdote but not an answer.
When I took abstract algebra (in 1956, way before categories) a question on the first exam asked for the definition of an isomorphism for an equivalence relation - not something we'd covered in class.
I (and most of my classmates) naively modified the definition we knew for group homomorphisms, requiring that the bijection $\phi$ satisfy $\phi(x) \equiv \phi(y)$ whenever $x \equiv y$. We all lost points.