Earlier this week, the UK Government published a paper outlining their offer for EU citizens in the UK, and UK nationals in the EU.
Further information about the offer for EU citizens can be found on GOV.UK: Status of EU citizens in the UK: what you need to know. The full policy paper can be read here: Safeguarding the position of EU citizens living in the UK and UK nationals living in the EU
As the paper suggests, EU nationals who are already living in the UK need not do anything now. The UK will remain a member of the EU until March 2019 and there will be no change to the rights and status of EU citizens living in the UK, nor UK nationals living in the EU, during this time. Therefore, EU citizens may not wish to apply for documentation confirming their status. Although, those EU nationals who are eligible to apply for permanent residence and/or British citizenship must make their application now, without any further delay.
The policy paper also sets out that the Home Office will be asking EU citizens to make an application for a residence document demonstrating their new settled status. They aim to make the process as streamlined and user-friendly as possible for all individuals, including those who already hold a permanent residence document under current free movement rules. They are expecting the new application system to be up and running in 2018.