  We present an introduction to the geometry of higher order vector and
co--vector bundles (including higher order generalizations of the Finsler
geometry and Kaluza--Klein gravity) and review the basic results on Clifford
and spinor structures on spaces with generic local anisotropy modeled by higher
order nonlinear connections. Geometric applications in locally anisotropic
gravity and matter field interactions are considered. This article contains a
part of material given by these authors and P. Stavrinos in theirs lectures.
