What are the main open problems in the theory of quasigroups and loops? 
What are the main open problems in the theory of quasigroups and loops?  

A short survey would be welcome. 
Thanks
 A: A list of problems can be found here:
http://en.wikipedia.org/wiki/Problems_in_loop_theory_and_quasigroup_theory
The question that attracted most attention over the past few years is, perhaps, "Loops with abelian inner mapping group". Various problems related to nilpotence and solvability seem to be alive.
(In my very personal opinion, one of the main open problems of loop theory is, to find out what are the main open problems of loop theory.)
A: I think one area that has seen a relatively recent major push is the classification of finite simple Bol loops, using techniques from the classification of finite simple groups.  The classification of finite simple Moufang loops is known (there is one infinite family of finite simple non-associative loops).  At the moment it seems that the full problem is too hard — it seems that there may be too many finite simple Bol loops to be classifiable — but there might be restricted classes that are classifiable.  Here are some slides from a presentation by Nagy that surveys the area.
