Summary: We examine the thermodynamic foundations of Cattaneo-Vernotte theory of heat conduction in solids. Some generalized theories of irreversible thermodynamics are shown to be unacceptable. Others are shown to be internally consistent but not completely satisfactory. A generalized definition of local temperature is given. Then, we propose a novel description of the thermodynamic states of a solid medium with a constant mass density and out of local equilibrium. According to this description, the generalized temperature fulfils a differential equation which, in general, contains time derivatives of order higher than the first. In particular, within this scheme, Fourier's law is compatible with a finite speed of propagation of thermal waves. The predictions of this new theory and those of the usual theory based on Cattaneo-Vernotte equation are compared through an example. It is shown that considerable differences between the temperature distributions predicted by the two theories occur when the heat flux density is specified at the boundaries.