  This article treats compactifications of the space of generalized spin
curves. Generalized spin curves, or $r$-spin curves, are pairs $(X,L)$ with $X$
a smooth curve and $L$ a line bundle whose r-th tensor power is isomorphic to
the canonical bundle of $X.$ These are a natural generalization of $2$-spin
curves (algebraic curves with a theta-characteristic), which have been of
interest recently, in part because of their applications to fermionic string
theory. Three different compactifications over $\Bbb{Z}[1/r],$ all using
torsion-free sheaves, are constructed. All three yield algebraic stacks, one of
which is shown to have Gorenstein singularities that can be described
explicitly, and one of which is smooth. All three compactifications generalize
constructions of Deligne and Cornalba done for the case when $r=2.$
