Specific heat is one of the oldest techniques used in condensed matter physics. This paper reviews the contributions made through specific heat measurements to our understanding of lattice vibrations in solids, starting with the work of Petit and Dulong in 1819. Recently, specific heat measurements have revealed new information on the lattice vibrations in glasses, i.e., structurally disordered systems and certain highly disordered, yet crystalline, solids. These two classes of solids contain anomalous low energy excitations which completely mask the specific heat resulting from plane wave excitations at low temperatures. Measurements of thermal relaxations (time dependent specific heat) in these solids have also demonstrated the limits of the concept of temperature in these solids in the low temperature limit.