Parametric surface waves arise spontaneously when a thin layer of granular material is oscillated vertically. Several models proposed at present account well for the observed instability threshold and the appearance of a variety of patterns but fail in connecting the peculiarities of granular materials with the observed pattern. In general, the proposed instability mechanism is due to two competing processes: a focusing effect that concentrates particles in space, and a diffusion effect that relaxes large thickness gradients in the layer. In these lectures, the dynamics of these waves is described and contrasted with the proposed theoretical models. When the layer is in a fluid type state, close to a solid-complex-fluid transition that we describe, localized waves, "oscillons", are observed. Achievements of theoretical models and open questions are briefly addressed.
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