  We construct regular integral canonical models for Shimura varieties attached
to Spin groups at (possibly ramified) odd primes. We exhibit these models as
schemes of 'relative PEL type' over integral canonical models of larger Spin
Shimura varieties with good reduction. Work of Vasiu-Zink then shows that the
classical Kuga-Satake construction extends over the integral model and that the
integral models we construct are canonical in a very precise sense. We also
construct good compactifications for our integral models. Our results have
applications to the Tate conjecture for K3 surfaces, as well as to Kudla's
program of relating intersection numbers of special cycles on orthogonal
Shimura varieties to Fourier coefficients of modular forms.
