Homogeneous dynamics: a study guide

These notes give a summary of the course ‘Ergodic theory and Ap- plications in Number Theory’ at the Summer School in Modern Mathe- matics at the Tsinghua University in Beijing, June 23–27 (2013). As in the Summer school, we will need to be brief at times and refer to the references for a detailed treatment. Nonetheless we wish to survey some results and ideas on a close to geodesic journey from the most basic con- cepts of ergodic theory to some more sophisticated and recent results in homogeneous dynamics.