  This paper is devoted to the horizontal (``characteristic'') cohomology of
systems of differential equations. Recent results on computing the horizontal
cohomology via the compatibility complex are generalized. New results on the
Vinogradov C-spectral sequence and the Krasil'shchik C-cohomology are obtained.
As an application of general theory, the examples of an evolution equation and
a p-form gauge theory are explicitly worked out.
