  We present a geometric approach to the field theory with higher order
anisotropic interactions. The concepts of higher order space, or locally
anisotropic, space (in brief, h-space, or la-space) are introduced as general
ones for various types of higher order extensions of Lagrange and Finsler
geometry and higher dimension (Kaluza-Klein type) spaces. The spinors on
h-spaces are defined in the framework of the geometry of Clifford bundles
provided with compatible nonlinear and distinguished connections and metric
structures (d-connection and d-metric). There are discussed some related issues
connected with the geometric aspects of higher order anisotropc interactions
for gravitational, gauge, spinor, Dirac spinor and Proca fields. The nearly
autoparallel maps are introduced as maps with deformation of connections
extending the class of geodesic and conformal transforms. We propose two
variants of solution of the problem of definition of conservation laws on
h-spaces. A general background of the theory of field interactions and strings
in spaces with higher order anisotropy is fromulated. The conditions for
consistent propagation of closed strings in higher order anisotropic background
spaces are analyzed. The connection between the conformal invariance, the
vanishing of the renormalization group beta-function of the generalized
sigma-model and field equations of higher order anisotropic gravity are studied
in detail.
