We review evidence of local dynamical lattice distortions in perovskite transition metal oxides, including cuprates and nickelates. These distortions appear as changes of the local symmetry around the metal ions, which are manifest in x-ray absorption spectroscopy and other inelastic scattering techniques. The distortions imply that these materials exhibit inhomogeneous ground states in some regions of their phase diagram. The distortions imply that these materials exhibit inhomogeneous ground states in some regions of their phase diagram. The presence of these distortions is a direct signature of polaronic excitations, as the Jahn-Teller polarons proposed by K.A. Müller, et al. [1] in the case of cuprates and that were previously studied by him in nickelates. [2] In the case of nickelates, we review direct evidence of an anomalous isotopic effect, characteristic of polaronic excitations as suggested by Professor Müller.