Kazhdan's property (T) is an important property for groups that stands at the roots of many rigidity results. Key examples of groups with property (T) are the groups SL(n,Z) with n > 2 (and more generally, higher rank lattices). A major application of property (T) is the explicit construction of expanders, which are sequences of finite sparse graphs with strong connectivity properties. Expanders play a large role in several areas of mathematics, including functional analysis, geometry and number theory. It is an important open problem whether expanders admit coarse embeddings into certain Banach spaces, i.e. embeddings that respect the large scale structure. After an elementary introduction to expanders and coarse embeddings, I will explain a joint work with Mikael de la Salle, in which we prove that given a Banach space satisfying certain mild conditions on its geometry, we can find an N such that for all n > N, the expanders constructed from SL(n,Z) (or a lattice of rank n) do not admit a coarse embedding into the given Banach space.