A representation of a group G is said to be rigid, if it cannot be continuously deformed to a non-isomorphic representation. If G happens to be the fundamental group of a complex projective manifold, rigid representations are conjectured (by Carlos Simpson) to be of geometric origin. In this talk I will outline the basic properties of rigid local systems and discuss several consequences of Simpson‘s conjecture. I will then outline recent progress on these questions (joint work with Hélène Esnault) and briefly mention applications to geometry and number theory such as the recent resolution of the André-Oort conjecture by Pila-Shankar-Tsimerman.