In January this year there was a new Settled Status scheme announced for EU citizens living in the UK. If you’re an EU citizen or a family member of an EU citizen, you’ll probably need to apply if you want to stay in the UK after 30 June 2021. The scheme is in a trial phase at the moment and will open fully in March 2019. The deadline for applying will be 30th June 2021 – unless there is a so-called no-deal Brexit in March, in which case the deadline for applying will be 31 December 2020.
If you are an employer with people working for you who are EU citizens, then you should do the usual checks on their immigration status. The scheme is quite complex – there is guidance on the Gov.uk website at https://www.gov.uk/settled-status-eu-citizens-families