TITLE: mapping space between classifying spaces
QUESTION [7 upvotes]: I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces when G and K are general topological groups. 
Thank you.

REPLY [10 votes]: The Sullivan conjecture, a beautiful theorem proved by Haynes Miller
(The Sullivan conjecture on maps from classifying spaces) has striking
consequences about maps between classifying spaces when the target is
completed at a prime $p$.  Some references are Dwyer and Zabrodsky
(Maps between classifying spaces), Jackowski, McClure, and Oliver
Homotopy classification of self-maps of $BG$ via $G$-actions. I, II;
Self-homotopy equivalences of classifying spaces of compact connected 
Lie groups); Notbohm (Maps between classifying spaces).  
The classical work of J.F. Adams on this topic is also well worth
remembering: Maps between classifying spaces (I), II, III and 
Maps between $p$-completed classifying spaces.