  We apply the theory of Borcherds products to calculate arithmetic volumes
(heights) of Shimura varieties of orthogonal type up to contributions from very
bad primes. The approach is analogous to the well-known computation of their
geometric volume by induction, using special cycles. A functorial theory of
integral models of toroidal compactifications of those varieties and a theory
of arithmetic Chern classes of integral automorphic vector bundles with
singular metrics are used. We obtain some evidence in the direction of Kudla's
conjectures on relations of heights of special cycles on these varieties to
special derivatives of Eisenstein series.
