Recently, I was wondering what would happen if one takes away the associativity property of groups, thereby stumbling across the concept of Moufang Loop. This seems to be one of the structures that comes closest to a group, even though it has a non-associative binary operation, so I figured a lot of group-theoretic results would have some sort of weaker analogs for Moufang loops. Then I read that the analog of something as "simple" as Lagrange's Theorem, that the order of every subloop of a finite Moufang loop divides the order of the original loop, was an open question for a long time and only proved recently.
So I was wondering, intuitively, what types of results I should expect to hold true for Moufang loops and what results I should expect to fail without full associativity? I'd also be very grateful if anyone knew of a good reference for an introductory but more or less thorough treatment of loop theory in general.