The Shafarevich conjecture (now a theorem of Arakelov and Parshin) is a statement concerning families of smooth algebraic curves.  It says that for any curve C, there are only finitely many admissible families of smooth curves of given genus over C, and that when such families exist, the base curve C has to be hyperbolic.  In this talk, I will introduce the relevant notions and explain the proof of the hyperbolicity of the base curve.  If time permits, I will also mention some higher-dimensional generalizations of the conjecture.