Developed by David Mumford in 1965, geometric invariant theory addresses the existence of quotients of schemes by group actions. I will set up the basic notions of stability and semi-stability, and then work out a few examples of the quotient construction (moduli of effective divisors on P^1, plane cubics). Time permitting, I will sketch Mumford's construction of Mg by applying G.I.T. to a certain Hilbert scheme.