Describing the behavior of orbits in a dynamical system is a fundamental question. In general one can only give information about the behavior of the orbit of a "typical" point. However, there are some ``rich" dynamical systems where extra structure of the system allows one to give rather precise information about orbits of individual points. Two important examples are unipotent flows on homogeneous spaces and the action of SL(2, R) on the moduli space of a compact Riemann surface.
In this talk we will give an overview of the developments around such rigidity results and discuss some applications.