We survey the geometry of Lagrange and Finsler spaces and discuss the issues
related to the definition of curvature of nonholonomic manifolds enabled with
nonlinear connection structure. It is proved that any commutative Riemannian
geometry (in general, any Riemann--Cartan space) defined by a generic
off--diagonal metric structure (with an additional affine connection possessing
nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler,
geometry modeled on nonholonomic manifolds. This results in the problem of
constructing noncommutative geometries with local anisotropy, in particular,
related to geometrization of classical and quantum mechanical and field
theories, even if we restrict our considerations only to commutative and
noncommutative Riemannian spaces. We elaborate a geometric approach to the
Clifford modules adapted to nonlinear connections, to the theory of spinors and
the Dirac operators on nonholonomic spaces and consider possible
generalizations to noncommutative geometry. We argue that any commutative
Riemann--Finsler geometry and generalizations my be derived from
noncommutative geometry by applying certain methods elaborated for Riemannian
spaces but extended to nonholonomic frame transforms and manifolds provided
with nonlinear connection structure.