The InSight (Interior Exploration using Seismic Investigations, Geodesy and Heat Transport) mission began collecting high quality seismic data on Mars from February 2019. This manuscript documents the seismicity observed by SEIS, InSight’s seismometer, since this time until the end of March, 2020. Within the InSight project, the Marsquake Service (MQS) is responsible for prompt review of all seismic data collected by InSight, detection of events that are likely to be of seismic origin, and curation and release of seismic catalogues. In the first year of data collection, MQS have identified 465 seismic events that we interpret to be from regional and teleseismic marsquakes. Seismic events are grouped into 2 different event families: the low frequency family are dominated by energy at long period below 1 second, and the high frequency family primarily include energy at and above 2.4~Hz. Event magnitudes, from Mars-specific scales, range from 1.3 to 3.7. A third class of events with very short duration but high frequency bursts have been observed 712 times. These are likely associated with a local source driven by thermal stresses. This paper describes the data collected so far in the mission and the procedures under which MQS operates; summarises the content of the current MQS seismic catalogue; and presents the key features of the events we have observed so far, using the largest events as examples.