Let $M$ be a compact Riemannian manifold on which a compact Lie group acts by isometries. In this talk I will explain how the symmetry induces extra structures in the spectrum of Laplace-type operators, and how to apply symplectic techniques to study the induced equivariant spectrum. In particular, I will discuss a) my joint works with V. Guillemin on inverse spectral results for Schrodinger operators on toric manifolds; b) my joint work with Y. Qin on the first equivariant eigenvalues of toric Kahler manifolds.