We present the results of a search for rapidly evolving transients in the Dark Energy Survey Supernova Programme. These events are characterized by fast light-curve evolution (rise to peak in ≲10 d and exponential decline in ≲30 d after peak). We discovered 72 events, including 37 transients with a spectroscopic redshift from host galaxy spectral features. The 37 events increase the total number of rapid optical transients by more than a factor of two. They are found at a wide range of redshifts (0.05 < z < 1.56) and peak brightnesses (-15.75 > Mg > -22.25). The multiband photometry is well fit by a blackbody up to few weeks after peak. The events appear to be hot (T ≈ 10 000–30 000 K) and large (R ≈ 1014 - 2 × 1015 cm) at peak, and generally expand and cool in time, though some events show evidence for a receding photosphere with roughly constant temperature. Spectra taken around peak are dominated by a blue featureless continuum consistent with hot, optically thick ejecta. We compare our events with a previously suggested physical scenario involving shock breakout in an optically thick wind surrounding a core-collapse supernova, we conclude that current models for such a scenario might need an additional power source to describe the exponential decline. We find that these transients tend to favour star-forming host galaxies, which could be consistent with a core-collapse origin. However, more detailed modelling of the light curves is necessary to determine their physical origin.