The UK government has published its agreement with the EU on how immigration of EU citizens will change post-Brexit. In short, it all depends on the date of your arrival and period of stay in the UK.
In order to ease the process, the UK government has agreed an implementation period from 30 March 2019 to 31 December 2020.
Your immigration status
All EU citizens arriving in the UK from 30 March 2019 must register to stay in the UK for over 3 months.
If you have stayed lawfully in the UK continuously for 5 years by 31 December 2020, you can apply for 'settled status' to stay indefinitely in the UK. If not, you can apply to remain until you complete 5 years of lawful stay. After this, you can apply for 'settled status' by 30 June 2021. Those with 'settled status' will have access to public funds and services.
From 1 July 2021, you must hold settled status or a temporary residence permit to stay in the UK.
Therefore, there is no requirement to do anything until 30 June 2021 if you are already in the UK before Brexit, unless you want to apply for British citizenship or sponsor your partner's visa application, in which case you will need a permanent residence document. Likewise, extended family member of EEA or Swiss nationals will also need a permanent residence document. Post-Brexit, the government has agreed that there will be a simple process to exchange your permanent residence document for a 'settled status' document free of charge.
If you have indefinite leave to remain, your rights will not affected by Brexit but you can exchange this for settled status free of charge to prove that you benefit from the withdrawal agreement.
Your family's immigration status
Your family members who join you in the UK by 31 December 2020 will be entitled to apply for settled status, usually after 5 years. Furthermore, anyone who is your close family member (spouse, civil and unmarried partners, dependent children and grandchildren, dependent parents and grandparents) on 31 December 2020 can join you in the UK post-Brexit.
Further immigration advice
Although the broad policy on the immigration status of EU nationals has been agreed, its details and the exact procedure that will be implemented and must be followed are still unknown. As the EU immigration policy is being re-written and UK immigration system is continuously changing, it will understandably be an uncertainty for many EU nationals currently living or hoping to move to the UK. There will be no, or very little, cases or word of mouth to go by. As such, it is crucial that you are aware of your options and the UK's new immigration requirements.