  We introduce the notion of even Clifford structures on Riemannian manifolds,
a framework generalizing almost Hermitian and quaternion-Hermitian geometries.
We give the complete classification of manifolds carrying parallel even
Clifford structures: K\"ahler, quaternion-K\"ahler and Riemannian products of
quaternion-K\"ahler manifolds, several classes of 8-dimensional manifolds,
families of real, complex and quaternionic Grassmannians, as well as
Rosenfeld's elliptic projective planes, which are symmetric spaces associated
to the exceptional simple Lie groups. As an application, we classify all
Riemannian manifolds whose metric is bundle-like along the curvature constancy
distribution, generalizing well-known results in Sasakian and 3-Sasakian
geometry.
