Using the Murchison Widefield Array (MWA), the low-frequency Square Kilometre Array (SKA1 LOW) precursor located in Western Australia, we have completed the GaLactic and Extragalactic All-sky MWA (GLEAM) survey, and present the resulting extragalactic catalogue, utilising the first year of observations. The catalogue covers 24,402 square degrees, over declinations south of +30{deg} and Galactic latitudes outside 10{deg} of the Galactic plane, excluding some areas such as the Magellanic Clouds. It contains 307,456 radio sources with 20 separate flux density measurements across 72-231MHz, selected from a time- and frequency- integrated image centred at 200MHz, with a resolution of ~=2'. Over the catalogued region, we estimate that the catalogue is 90% complete at 170mJy, and 50% complete at 55mJy, and large areas are complete at even lower flux density levels. Its reliability is 99.97% above the detection threshold of 5{sigma}, which itself is typically 50mJy. These observations constitute the widest fractional bandwidth and largest sky area survey at radio frequencies to date, and calibrate the low frequency flux density scale of the southern sky to better than 10%. This paper presents details of the flagging, imaging, mosaicking, and source extraction/characterisation, as well as estimates of the completeness and reliability. All source measurements and images are available online. This is the first in a series of publications describing the GLEAM survey results.