The talk will report on work with Ron Guenther. It is part of numerical analysis folklore that eigenvalue problems for nice ordinary differential equations can be solved numerically using a shooting approach. What is missing from the folklore is the application of this approach to an interesting class of problems supported by a convergence analysis. We provide such an approach for both regular and singular Sturm-Liouville problems. In the talk, I'll explain how things work for regular problems and indicate what is needed to accommodate singular problems. Singular here means we consider singularites that occur when wave and heat equations are solved by separation of variables for problems that involve circular, cylindrical, or spherical symmetry of some sort. Numerical examples will be given for both regular and singular problems. If you know what an eigenvalue problem is and what an initial value problem for an ordinary differential equation is you can understand this talk.