  We discuss three recent developments in the calculation of CMB anisotropies.
First, we consider relativistic corrections to the Sunyaev-Zel'dovich effect.
By extending the Kompaneets equation to the relativistic domain we are able to
derive simple analytic forms for the SZ spectral distortion. Second, we discuss
covariant kinetic theory methods for the gauge-invariant calculation of CMB
anisotropies. We present a covariant version of the Boltzmann equation which we
use to derive covariant and gauge-invariant equations describing a CDM model,
which are independent of background curvature and type of perturbation (scalar,
vector or tensor). By integrating the covariant Boltzmann equation along the
line of sight, and using the instantaneous recombination approximation, we
obtain an expression for the large-scale anisotropy which agrees with an
expression derived recently by Dunsby. Third, we discuss recent accurate
calculations of the CMB anisotropy in global defect models of cosmic structure
formation. In such models the effects of vector modes are significant and when
combined with the effects of decoherence lead to a suppression of the Doppler
peaks. The result is that global defect models may already be at variance with
the growing volume of CMB and large scale structure data.
