What is...?

Harmonic functions on groups are connected to many properties of the groups: algebraic, geometric, analytic, and probabilistic.
For some groups (or spaces), it can be a challenge even to determine whether harmonic functions of certain types—such as bounded or positive—exist.
The task is then to describe all such functions, understand their properties, and study the properties of the group action on them. Other classes of functions, such as Lipschitz functions, have been described more recently and shown to have striking geometric applications. I will sketch some of the results in this area and explain several open questions.