Let us use this answer to bring additional relevant links.

Here is a blog post [Caterorigication step I][1] by Peter Cameron based on a lecture by Igor Frenkel.

I myself heard an amazing lecture by David Kazhdan last year were the primary example was the step of moving from characters to representation. 

  [1]: http://cameroncounts.wordpress.com/2014/06/12/categorification-step-1/