What are some of the major open problems in category theory?

What are some of the major open problems in category theory? Just curious - I'm interested in category theory.

It depends really on what you mean by open problem in category theory. By that I mean that one must make a clear distinction between research that employs categorical language, and research of category theory. The reason is very simple: category theory is a very widely used language for discussing mathematics. This is much like the situation with set theory. Set theory is the common-place language we use to discuss mathematics. Virtually any piece of mathematics would contain some elements of set theory, simply as the vocabulary making up the language. Does that mean that any research problem belongs to Set Theory? Of course not! That would be like saying that the research problem "What exactly did Napoleon eat for breakfast on the day of his death?" (which is a research problem in nonsenseological recent history as well as in nonsenseological political culinary studies) belongs to the study of the English language. Now that is nonsense.

So, research problems that heavily use category theory language and machinery may include such things as the cobordism hypothesis and its variants. Other such problems would include problems in the theory of model categories (i.e., homotopical algebra). Hovey's book "Model categories" has a rather long list of open problems in the area, some of which were already solved but many remain open. Research problems that are more purely category theoretic (though they are motivated by applications, of course) must include the very active area of research known as higher category theory. This includes the immense work of Lurie on higher topos theory as well as that of many other contributors working on unifying ideas in higher category theory. The nLab is a very good place to read on these issues.

One can also mention the recent work on homotopy type theory - a new foundations of mathematics. It contains aspects of both types of research problems mentioned above.

I hope this gives you a place to start looking for more concrete information.