The SkyMapper Transient survey (SMT) is exploring variability in the southern sky by performing (a) a rolling search to discover and study supernovæ, and (b) a Target of Opportunity programme that uses the robotic SkyMapper Telescope at Siding Spring Observatory. The supernova survey is obtaining a non-targeted sample of Type Ia supernovæ (SNe Ia) at low redshifts, z < 0.1, and studying other interesting transients found with the search strategy. We have a Target of Opportunity programme with an automatic response mechanism to search for optical counterparts to gravitational-wave and fast radio-burst events; it benefits from SkyMapper's large field of view of 5.7 sq. deg. and a rapid data reduction pipeline.We present first results of the SMT survey. The SMT pipeline can process and obtain potential candidates within 12 hours of observation. It disentangles real transients from processing artefacts using a machine-learning algorithm. To date, SMT has discovered over 60 spectroscopically confirmed supernovæ, several peculiar objects, and over 40 SNe Ia including one (SNIa 2016hhd) which was found within the first few days of explosion. We have also participated in searches for optical counterparts of gravitational waves, fast radio bursts and other transients, and have published observations of the optical counterpart of the gravitational-wave event GW170817. We also participate in coordinated observations with the Deeper Wider Faster programme, and the Kepler K2 cosmology project.