Abstract: We give a survey of what is known about how many symmetries an algebraic variety can possess. We start with some classical results, including those of Hurwitz, Noether and Riemann, to do with the automorphism group of the plane and the automorphism group of curves (or equivalently Riemann surfaces), and we end with some more recent results to do with the automorphism group of threefolds of low degree and varieties with finite automorphism group.