The topic to be discussed here is how we can use the Cosmic Microwave Background to set constraints on the Hubble Constant. Up to now, this has meant almost exclusively the use of the Sunyaev-Zel'dovich effect in clusters of galaxies, which will be the initial focus of this review. This use of SZ effect is important as a physical method which bypasses the usual distance ladder arguments for establishing H0. However, a recent additional method, which holds very great promise for the future, is to use the power spectrum of primordial microwave anisotropies to constrain H0, and this will be discussed in the final section, together with some first results in this direction.
In Section 2 we start by discussing the Sunyaev-Zel'dovich (SZ) effect, what it is, and current observations of it. Then we review the procedure of obtaining H0 from SZ measurements in conjunction with X-ray information, and discuss the current status of the results, particularly in connection with systematic errors. A key feature of the SZ effect is its redshift independence. This will be illustrated by the interesting example of the discovery of a microwave decrement in front of a quasar pair at a redshift of z 3.8. Finally, we discuss how H0 can be determined from the power spectrum of primordial fluctuations of the CMB, and argue that it is in fact via this route that H0 will eventually be pinned down to high accuracy.