TITLE: Where to study $2$-category theory?
QUESTION [9 upvotes]: Is there any place where I can read about $2$-categories? I am looking for a proper treatment - there is a section in Borceux's Handbook of Categorical Algebra, but it only sketches some parts of the theory, and it does not include any proofs. If it helps, I would like to get familiar with what is needed in order to be able to deal with the $2$-categorical structures involved in topos theory. 
Is it just me, or does anyone else think that category theory is developing quicker than it can be written down? ... I have the same problem with internal category theory.
Any help would be highly appreciated.

REPLY [6 votes]: I think the combination of Basic Bicategories by Tom Leinster for basic definitions, and A 2-categories companion by Stephen Lack for the essential 2-categorical constructions is a good way to get a feel for the theory, although the latter is a fairly informal and nonrigorous account.

REPLY [3 votes]: 2-categories have been considered for nearly as long as categories. I think it's more that the field isn't large enough to support an exhaustive range of advanced textbooks, and that the natural progression is either to study a specific 2-category or to go on to $n$- and $\infty$-categories. That said, I believe you can find everything you need for toposes in Johnstone's Sketches of an Elephant, and you might also look at Amnon Neeman's monograph on triangulated categories. Kelly and Street, "Review of the Elements of 2-Categories" comes highly recommended, but I've never seen it.