TITLE: Uniqueness of loop spaces
QUESTION [7 upvotes]: Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$. 
Under what assumptions is (the homotopy type of) $Y$ unique? 
As has been pointed out below, the homotopy type of $Y$ being determined uniquely is far from true in general. But for connected $Y$, are there conditions we can impose that make it so?

REPLY [12 votes]: As others have pointed out, the generic case (whatever that should mean in this case) is that the loop structure on a loop space is not unique. However, things get quite interesting whenever we have a space that actually does have a unique loop structure. I highly recommend looking at:
Dwyer, Miller, Wilkerson: The homotopic uniqueness of $BS^3$, LNM 1298
and
Dwyer, Miller, Wilkerson: Homotopical uniqueness of classifying spaces. Topology 31 (1992), no. 1, 29–45.