The isoperimetric problem on a Riemann surface seeks a region of least perimeter enclosing a given area. On the plane, for example, the isoperimetric region is a circle. However, despite its relatively simple formulation, many variations of this problem are still open. In this talk I will give a brief introduction to the Riemannian geometry of surfaces, and discuss the isoperimetric problem in the case where the surface is weighted with a density function.