The UK is asking European residents there to apply for a special status, but the other 27 have not yet decided what UK residents in their countries will do.
We could apply for nationality; we could (in France) apply for a carte de sejour, possibly for permanent status if resident here for more than 5 years.
Different prefectures ask for different documents, but proof of residence for each year, proof of EU citizenship (eg passport), proof of health insurance (could be a carte vitale). If there is a "no deal" decision we will have no particular rights, so it could be important to make an application to your prefecture asap.
An association ECREU, dedicated to keeping as many rights as possible if the "leave" decision is carried through, has posted a useful questionnaire at https://goo.gl/6wNyK4.
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