My research is mostly about exploring how symmetries can be used to generate randomness or unveil structural insights. In this talk, I will focus on random walks on compact groups, and give you a glimpse of some of the tools that I use to study such a random process:

• Connection with expander graphs,
• Property (T),
• Growth within algebraic structures: sum-product and product results,
• Entropy and the Bourgain-Gamburd technique.

At the end, I will mention more recent results of Srinivas and mine on random walks on group extensions.