This article treats the compactification of the space of higher spin curves, i.e. pairs (X,L) with L an r root of the canonical bundle of X. More precisely, for positive integers r and g, with g > 2, r dividing 2g − 2, and for a flat family of smooth curves f : X → T, an r-spin structure on X is a line bundle L such that L ∼= ωX /T . And an r-spin curve over T is a flat family of smooth curves ...