  We present a fully covariant and gauge-invariant calculation of the evolution
of anisotropies in the cosmic microwave background (CMB) radiation. We use the
physically appealing covariant approach to cosmological perturbations, which
ensures that all variables are gauge-invariant and have a clear physical
interpretation. We derive the complete set of frame-independent, linearised
equations describing the (Boltzmann) evolution of anisotropy and inhomogeneity
in an almost Friedmann-Robertson-Walker (FRW) cold dark matter (CDM) universe.
These equations include the contributions of scalar, vector and tensor modes in
a unified manner. Frame-independent equations for scalar and tensor
perturbations, which are valid for any value of the background curvature, are
obtained straightforwardly from the complete set of equations. We discuss the
scalar equations in detail, including the integral solution and relation with
the line of sight approach, analytic solutions in the early radiation dominated
era, and the numerical solution in the standard CDM model. Our results confirm
those obtained by other groups, who have worked carefully with non-covariant
methods in specific gauges, but are derived here in a completely transparent
fashion.
