There has recently been activity in the area of operators diagonalizable or almost diagonalizable by generalizations of bases called frames, whether it is with the theory of Gabor or frame multipliers or the almost diagonalization of pseudodifferential operators by Gabor frames. In this talk, we explore the concept of frame diagonalization in both finite and infinite dimensions and investigate the properties of classes of operators whose matrix representation with respect to a pair of frames is almost diagonal.