  The paper contains a review on recent progress in the deformational
properties of smooth maps from compact surfaces $M$ to a one-dimensional
manifold $P$. It covers description of homotopy types of stabilizers and orbits
of a large class of smooth functions on surfaces obtained by the author, E.
Kudryavtseva, B. Feshchenko, I. Kuznietsova, Yu. Soroka, A. Kravchenko. We also
present here a new direct proof of the fact that for generic Morse maps the
connected components their orbits are homotopy equivalent to finite products of
circles.
