Abstract. I'll discuss a joint work with Bader, Furman and Monod. We studied Kazhdan property (T) and the fixed point property for actions on Banach spaces instead of Hilbert spaces. We showed that property $(T)$ holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We showed that the fixed point property for $L^p$ follows from property (T) when $1 < p < 2+\epsilon$. For simple Lie groups and their lattices, we proved that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtained a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces. Some of our results have applications to the study of group actions on manifolds.