**Synopsis:**
One way of understanding groups is by investigating their actions on special spaces,
such as Hilbert and Banach spaces, non-positively curved spaces etc.

Classical properties such as Kazhdan's property (T) and the Haagerup property are formulated in terms
of such actions and turn out to be relevant in a wide range of areas, e.g. combinatorics, algebra, dynamics and
in relation to the conjectures of Novikov and Baum-Connes.

In this course I shall overview property (T) and Haagerup, and their various generalisations to Banach spaces,
especially in connection with classes of groups acting on non-positively curved spaces.

**Lecture notes: **