  Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves
have been generalized to the case of generic planar wave fronts. We generalize
these invariants to the case of wave fronts on an arbitrary surface $F$. All
invariants satisfying the axioms which naturally generalize the axioms used by
Arnold are explicitly described. We also give an explicit formula for the
finest order one $J^+$-type invariant of fronts on an orientable surface $F\neq
S^2$. We obtain necessary and sufficient conditions for an invariant of
nongeneric fronts with one nongeneric singular point to be the Vassiliev-type
derivative of an invariant of generic fronts. As a byproduct, we calculate all
homotopy groups of the space of Legendrian immersions of $S^1$ into the
spherical cotangent bundle of a surface.
