A finite loop space is a pointed space BG such that the loop space ΩBG is homotopy equivalent to a finite CW-complex. The classification of connected p-compact groups implies a classification of connected finite loop spaces : Given a connected p-compact group for each prime, all with the same rational type, there is an explicit double coset space of possible connected finite loop spaces with p-completion the give p-compact groups. As connected p-compact groups are classified combinatorially, this implies a classification of connected loop spaces as well.