The fundamental group π1(X) is an important invariant of a complex algebraic variety X.  Despite its topological nature, it is closely connected to the geometry of many algebraic structures on X.  In this talk I want to discuss two elementary questions involving the fundamental group:(1) How many simply-connected algebraic subvarieties does X contain?(2) How many global holomorphic functions does the universal cover X̃have?The second question is related to a famous conjecture of Shafarevich, which asserts that in the case X is a smooth projective variety, X̃has enough global holomorphic functions to separate points up to compact ambiguity.  I will describe joint work with Brunebarbe and Tsimerman which gives a complete answer to questions (1) and (2) provided π1(X)admits a faithful linear representation, using techniques coming from non-abelian Hodge theory.