Despite considerable efforts over the past decade, only 34 fast radio bursts-intense bursts of radio emission from beyond our Galaxy-have been reported(1,2). Attempts to understand the population as a whole have been hindered by the highly heterogeneous nature of the searches, which have been conducted with telescopes of different sensitivities, at a range of radio frequencies, and in environments corrupted by different levels of radio-frequency interference from human activity. Searches have been further complicated by uncertain burst positions and brightnesses-a consequence of the transient nature of the sources and the poor angular resolution of the detecting instruments. The discovery of repeating bursts from one source(3), and its subsequent localization(4) to a dwarf galaxy at a distance of 3.7 billion light years, confirmed that the population of fast radio bursts is located at cosmological distances. However, the nature of the emission remains elusive. Here we report a well controlled, wide-field radio survey for these bursts. We found 20, none of which repeated during follow-up observations between 185-1,097 hours after the initial detections. The sample includes both the nearest and the most energetic bursts detected so far. The survey demonstrates that there is a relationship between burst dispersion and brightness and that the high-fluence bursts are the nearby analogues of the more distant events found in higher-sensitivity, narrower-field surveys(5).