TITLE: Motivation and use for category theory?
QUESTION [36 upvotes]: From reading the answers to different questions on category theory, it seems that category theory is useful as a framework for thinking about mathematics. Also, from the book Algebra by Saunders Mac Lane, in the preface for first edition there is a passage, 
"... It is now clear that we study not just a single algebraic system (a group or a ring) by itself, but that we also study the homomorphisms of these systems; that is, the functions mapping one system into another as as to preserve the operations... All the systems of a given type together with the homomorphisms between them are said to form a "category" consisting of "objects" and "morphisms"..." This book proposes to present algebra for undergraduates on the basis of these new insights." 

This quote alone probably means that category theory is worth studying from a mathematician's perspective. But I would like to see examples of it being used to solve more concrete problems. I have read in Miles Reid's Commutative Algebra that Grothendieck  used category theory successfully to solve problems in Algebraic geometry while at the same time saying that category theory is one of the most sterile intellectual pursuit for most students. If this is true then the example will be hard to understand, perhaps someone here could provide a non-trivial yet accessible example? Edit: Personally I have read somewhere (probably n-Category cafe) that its possible to construct groups entirely from categories and functors, and I really liked this idea because it illustrates how to think in terms of categories really well. It would be nice to see more examples like these.

P.S. for background, I am in the process of reading most of Saunder Mac Lane's books. I have also read and in the process of reading a few of Emil Artin's book on algebra as well (e.g. Galois theory, Algebra and Galois theory and Geometric Algebra). Also if this post is more appropriate for Community Wiki then please change it for me.

REPLY [13 votes]: I had the idea when I was in grad school that category theory was a convenient language in which to state things, but didn't have any deep results. However, I now believe nothing could be further from the truth! Category theory has now shown itself to be incredibly useful in topology for very concrete problems, such as finding knot invariants and even 3 and 4-manifold invariants. Reshetikhin-Turaev invariants, the Kontsevich integral, and Khovanov homology are powerful link invariants that all arise via a categorical approach.