After 25 years, the microscopic mechanism of High-Tc superconductivity is still an open problem. In particular, the role of lattice vibrations is not yet clear. Many of the proposed models invoke unusual vibrational properties, such as nonlinearity in the interatomic potentials. In this thesis we present a combined experimental and theoretical study of lattice anharmonicities in cuprates. We first introduce and describe our Extended X-Ray Absorption Spectroscopy (EXAFS) experiments. We then develop a novel data analysis technique, which enables us to measure directly the nonlinearity in the interatomic potentials of the copper-oxide planes, which constitute the basic structural unit of all cuprates. Finally, based on our experimental results, we introduce a simple dynamical model of the copper-oxide layers. We study this model both analytically and numerically, showing the emergence of anharmonic localised excitations known from nonlinear physics as discrete breathers.