The Hilbert scheme of points on a quasi-projective variety parameterizes its zero-dimensional subschemes. These Hilbert schemes are smooth and irreducible for smooth surfaces but will eventually become reducible for sufficiently singular surfaces. In this talk, I provide the first class of examples of singular surfaces whose Hilbert schemes of points are irreducible, namely surfaces with at worst cyclic quotient rational double points. I will also describe some consequent geometric properties of these irreducible Hilbert schemes.