The Withdrawal Agreement between the Europan Union and the United Kingdom came into effect on 1 February 2020. On that day, the UK ceased to be a member of the EU. There is a transitional period in place until 31 December 2020, during which EU law will continue to apply to the UK. Therefore, during this transitional period, there will be no legal changes for the British citizens living in the member states of the European Union nor for EU citizens living in the UK.
You can find detailed information on the impact of Brexit after the transitional period on the website of the Federal Chancellery, in particular on: