  An introduction into the theory of locally anisotropic spaces (modelled as
vector bundles provided with compatible nonlinear and distinguished linear
connections and metric structures and containing as particular cases different
types of Kaluza--Klein and/or extensions of Lagrange and Finsler spaces) is
presented. The conditions for consistent propagation of closed strings in
locally 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 locally anisotropic
gravity is studied in detail.
