Another link than the one explained by Charles Stewart is the relation with accessible categories. This was proposed by Michael Makkai and Bob Paré as a category theoretic foundation for model theory (Accessible categories: the foundations of categorial model theory, Contemporary Mathematics 104, AMS, 1989). I found this approach particularly compelling.
The basic idea is to think of the of models of a complete theory as forming a category with elementary embeddings as morphisms. The fact that this is an accessible category is basically the Löwenheim-Skolem Theorem. I really like the fact that this view is not limited by first-order logic. For example, it applies to infinitary logics and Abstract Elementary Classes just as well.
Another connection comes though classifying topoi (see McLane Mac Lane & Moerdijk, Sheaves in Geometry and Logic), Chapter X). There are also strong ties with Abstract Stone Duality (I'm still trying to catch up there, so I can't say much more).