The giant molecular clouds (MCs) found in the Milky Way and similar galaxies play a crucial role in the evolution of these systems. The supernova explosions that mark the death of massive stars in these regions often lead to interactions between the supernova remnants (SNRs) and the clouds. These interactions have a profound effect on our understanding of SNRs. Shocks in SNRs should be capable of accelerating particles to cosmic ray (CR) energies with efficiencies high enough to power Galactic CRs. X-ray and gamma-ray studies have established the presence of relativistic electrons and protons is some SNRs and provided strong evidence for diffusive shock acceleration as the primary acceleration mechanism, including strongly amplified magnetic fields, temperature and ionization effects on the shock-heated plasmas, and modifications to the dynamical evolution of some systems. Because protons dominate the overall energetics of the CRs, it is crucial to understand this hadronic component even though electrons are much more efficient radiators and it can be difficult to identify the hadronic component. However, near MCs the densities are sufficiently high to allow the gamma-ray emission to be dominated by protons. Thus, these interaction sites provide some of our best opportunities to constrain the overall energetics of these particle accelerators. Here we summarize some key properties of interactions between SNRs and MCs, with an emphasis on recent X-ray and gamma-ray studies that are providing important constraints on our understanding of cosmic rays in our Galaxy.