The proof of Fermat's Last Theorem established a deep relation between elliptic curves and modular forms, mediated by an equality of L-functions (which are analogous to the Riemann zeta function). The common ground is Galois representations, and Wiles' overall strategy was to parametrize their deformations via algebras of Hecke operators. In higher rank the global Langlands conjecture posits a correspondence between n-dimensional Galois representations arising from the cohomology of algebraic varieties and certain so-called automorphic representations of $GL(n)$, which belong in the realm of harmonic analysis. There is a known analogue over local fields (such as the p-adic numbers $Q_p$) and one of the key desiderata is local-global compatibility. This naturally leads one to speculate about the existence of a finer "p-adic" version of the local Langlands correspondence which should somehow be built from a "mod p" version through deformation theory. Over the last decade this picture has been completed for $GL(2)$ over $Q_p$, and extending it to other groups is a very active research area. In my talk I will try to motivate these ideas, and eventually focus on deformations of smooth representations of $GL(n)$ over $Q_p$ (or any p-adic reductive group). It seems to be an open problem whether universal deformation rings are Noetherian in this context. At the end we report on progress in this direction (joint with Julien Hauseux and Tobias Schmidt). The talk only assumes familiarity with basic notions in algebraic number theory.