Type Theory has for a while had the potential to replace (Zermelo Fraenkel) set theory as a foundation for mathematics. It allows to manipulate proofs just like programs which makes Type Theory especially attractive for automated theorem checking or -proving and other such applications and it even potentially enables programmers to mathematically prove bug-freeness of at least certain classes of bugs: No thread-blocking-issues, no memory-leaks, no out-of-bound arrays... These things can become provable using Type Theory.
Fairly recently, the concept of Types has been associated with the concept of Homotopy which is a certain topological notion. Two geometric structures are homotopically equal, if there exists a continuous function that maps one object into the other. (There is the stronger notion of Isotopy which requires that such a transformation does not have any points of the geometric objects intersect with each other during the transformation, so for instance, the intervals [a,b] and [b,a] are not isotopic but they are homotopic by the continuous function in t: [a (1-t)+b t, b(1-t) + a t] - note that the end points swap places and so must, at some point, pass through each other. - Namely at t=1/2)
Through concepts from Homotopy, you gain several interesting things. Two of which are:
For one, you end up with a certain kind of equality which makes all isomorphic structures equal to each other (something that is highly wanted, since you don't typically care in which way, say, the integers are defined. As long as all definitions share certain properties, integers will still be integers. But if you don't have isomorphisms as equal, this will not be obvious.)
And for another you gain some notion of "higher dimensional types" which are notions that, as far as I understood it, would be almost impossible to emulate in set-theory.
A simple type "A" would be a "point"-like object. A type "A -> B" would be somewhat like a line. A type (A->B)->(C->D) would be equivalent to a plane. - at least to my current understanding. I'm pretty sure, I'm grossly oversimplifying here.
Anyway, last year was entirely dedicated to further the development of Homotopy Type Theory and amongst many fruitful papers for once, several of the people working in this field have teamed up to also release a fully freely available (online) book (you can also pay for a printed version if you so choose)
http://homotopytypetheory.org/book/and a carnegie mellon lecture on the topic
http://scs.hosted.panopto.com/Panopto/Pages/Sessions/List.aspx#folderID=%2207756bb0-b872-4a4a-95b1-b77ad206dab3%22the styles of the book and the lecture differ quite a bit. The book starts directly with type theory and develops all else from there. The lecture starts with logic and highlighting the differences between "classical" logic and the kind of logic you get through this system.