  We investigate cosmological density perturbations in a covariant and
gauge-invariant formalism, incorporating relativistic causal thermodynamics to
give a self-consistent description. The gradient of density inhomogeneities
splits covariantly into a scalar part, equivalent to the usual density
perturbations, a rotational vector part that is determined by the vorticity,
and a tensor part that describes the shape. We give the evolution equations for
these parts in the general dissipative case. Causal thermodynamics gives
evolution equations for viscous stress and heat flux, which are coupled to the
density perturbation equation and to the entropy and temperature perturbation
equations. We give the full coupled system in the general dissipative case, and
simplify the system in certain cases. A companion paper uses the general
formalism to analyze damping of density perturbations before last scattering.
