In this talk, I will explain a method that, for any N>0, produces a sequence of smooth manifolds and a smooth orbifold having nontrivial singular set such that the first N eigenvalues of the Laplace spectra of the manifolds come arbitrarily close to the first N eigenvalues of the orbifold. Here, the result holds whether the Laplacian acts on functions or on forms. I will also explain how for any N>0, it is possible to prescribe the first N eigenvalues of an oriented orbifold of dimension greater than or equal to 3. These results are achieved by generalizing results of Colin de Verdière, Jammes, and Rauch and Taylor to the orbifold setting, and by proving a Hodge decomposition theorem for orbifolds having manifold boundary. This is joint work with Carla Farsi and Christopher Seaton.