Ergodic theory studies dynamical systems and group actions in the presence of measures. Typical statements in this field describe existence (or lack) of measurable maps with certain properties, distribution of orbits of typical points etc. It is surprising and fascinating that such phenomena turn out to be very useful in proving some rigidity results in geometry. In this mini-course I will mention a few recent results related to arithmeticity and linearity, and then focus on a rigidity problem involving delta-hyperbolic spaces. Most of the material is based on joint works with Uri Bader.