The use of Type Ia supernovae as standard candles led to the discovery that the expansion of the universe is accelerating. The Supernova Cosmology Project High Redshift Supernova Survey aims to improve the constraint on the dark energy equation of state above z ~ 1 by a factor of 3, allowing us to probe the time variation of dark energy. We were awarded 174 orbits of HST time over two years (ending autumn 2016) to regularly observe massive clusters between z ~ 1 and z ~ 1.8, with the aim of discovering Type Ia supernovae. These cadenced visits, along with ToO observations of active supernovae, allow us to construct a multi-colour light curve for each supernova. The HST programme is supported by a large ground-based spectroscopic follow-up programme aimed at obtaining supernova classification and redshifts so the Type Ia supernovae can be used for cosmology. This includes VLT, Keck, Gemini and GTC, and in some cases light from the supernovae themselves may be detected in the spectra. In the majority of cases, where a direct supernova spectrum is not possible, an accurate redshift from the narrow emission/absorption lines of the host galaxy, combined with the HST multi-colour light curve can give a strong supernova type indication. Here we give an overview of the survey and the progress to date.