Is there a relationship between model theory and category theory?

Categorical logic is definitely one place to look. Although in my very limited experience model theory seems to be mostly interested in models built out of sets, while categorical logic is usually primarily interested in models in more general categories.

Another place to look is the theory of locally presentable and accessible categories, which are the categories of models (in Set) of a large class of theories. A number of model-theoretic ideas and techniques come up in their study, and the converse might be true too. The standard books are:

• Adámek and Rosicky, Locally presentable and accessible categories
• Makkai and Paré, Accessible categories: the foundations of categorical model theory

(Makkai and Paré certainly thought there was a relationship!)

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 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).

There certainly is. Last year there was a largeish conference in Durham, New Directions in the Model Theory of Fields, which had the connections between model theory and category theory as its "second theme".

Perhaps the talk most relevant to your question was that of Martin Hyland, Categorical Model Theory. You can see a video on the website, but unfortunately it seems to start part-way through the talk. Anyway, he started by saying that everything he was going to explain was known in 1982, which perhaps was a reference to Makkai-Pare (as mentioned by Mike Shulman and F.G. Dorais) and that era.

A distinguished, but non-categorical, logician who seems to strongly support categorical model theory is Angus Macintyre. Here's his introduction to 'Model theory: geometrical and set-theoretic aspects and prospects', Bulletin of Symbolic Logic 9 (2003):

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older "sets of points in affined or projective space" no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and "transfer of structure" is incomparably more flexible and powerful then anything yet known in "set-theoretic model theory".

It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called "Definability Theory" in the near future.