A system is considered to be an elastic solid if its constituent particles are in mechanical equilibrium, and, at least on the linear level, the energy (or the free energy, as in the case of colloidal solids) grows under any displacement of the particles from their equilibrium positions. These seamingly simple conditions actually encapsulate deep and interesting physics. For example, consider the simplest model for a solid: point masses connected with springs; if the mean number of springs per mass is smaller than twice the dimension, there will exist non-trivial displacements of the masses that cost no energy at all - a direct violation of the solidity condition. As more interactions between the masses are considered, the number of independent zero energy displacements starts to decrease, until they disappear completely when the number of interactions reaches exactly twice the spatial dimension. The disappearance of these "soft" displacements is known as "jamming", and is believed to constitute a non-equilibrium critical point, with associated diverging correlation lengths and scaling properties. In my research I employ simple models of solids to study the physics of this critical point, and its implications for the dynamics, mechanics and flow of more realistic elastic solids.