Efficient and accurate simulations of wave propagation have a wide range of applications in science and engineering, from seismic and medical imaging to rupture and earthquake simulations. In this thesis, we focus on provably energy stable, efficient and high order accurate discontinuous Galerkin (DG) methods for wave problems with general choices of basis and quadrature. We first extend existing DG solvers for the acoustic and elastic wave equations to coupled elastic-acoustic media. A simple upwind-like numerical flux is derived to weakly impose continuity of the normal velocity and traction at elastic-acoustic interfaces. When paired with the weight-adjusted DG (WADG) method, the resulting scheme is consistent and energy stable on curvilinear meshes and for arbitrary heterogeneous media, including anisotropy and sub- cell heterogeneities. We also present an application of this coupled DG solver to an inverse problem in photoacoustic tomography (PAT). Then, we develop DG methods for wave equations on moving meshes. In the proposed method, an arbitrary Lagrangian-Eulerian (ALE) formulation is adopted to map the acoustic wave equation from the time-dependent moving physical domain to a fixed reference domain. The spatially varying geometric terms produced by the ALE transformation are approximated using a weight-adjusted DG discretization. The resulting semi-discrete WADG scheme is provably energy stable up to a term which converges to zero with the same rate as the optimal L2 error estimate. Finally, two mesh regularization approaches are proposed to address the possible mesh tangling caused by boundary deformations. We use the Bernstein triangular representation to convert a curved mesh into the Bernstein control net which only consists of linear sub-triangles. The validity of curved meshes is guaranteed when all sub-triangles are untangled. Our first approach borrows the idea of the spring-mass system for adaptation of first-order linear meshes. The second approach solves an optimization problem to prevent the appearance of small Jacobian determinants of each sub-triangle in the control net.