Various dynamical systems in dimensions one and two (like interval exchange transformations, billiards in rational polygons, measured foliations on surfaces) lead naturally to dynamics along straight lines (geodesics) on ``flat surfaces'': compact surfaces equipped with a flat matric having several specific cone type singularities. Such flat structure is equivalent to a pair: complex structure plus a quadratic differential. Thus, families of flat surfaces correspond to the moduli spaces of quadratic differentials.

It happens that the Teichmüller geodesics flow acting on the moduli space of quadratic differentials plays a role of a renormalisation procedure for the original dynamical system. In many cases some complicated properties of "simple" dynamics on a generic flat surface can be related to simple properties of more complicated dynamics on the moduli space.

I want to suggest an elementary introduction to the subject, to give a short report on recent advances in this area, and to present briefly several key problems in Teichmuller dynamics.