We consider the theory of functions of bounded variation (BV functions) in the general setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Such a theory was first developed by Ambrosio (2002) and Miranda (2003). I will give an overview of the basic theory and then discuss a metric space proof of Federer's characterization of sets of finite perimeter, i.e. sets whose characteristic functions are BV functions. This characterization states that a set is of finite perimeter if and only if the n-1-dimensional (in metric spaces, codimension one) Hausdorff measure of the set's measure-theoretic boundary is finite. The proof relies on fine potential theory in the case p=1, much of which seems to be new even in Euclidean spaces.