Can I apply for British citizenship and still keep my EU passport / nationality?

It depends on your EU country of citizenship. 

Some countries, such as The Netherlands, Austria, Spain, Estonia, Lithuania, Latvia, Slovakia, Slovenia and Germany, have restrictions on dual citizenship. 

Others, such as Romania, allow dual citizenship with the UK for all their citizens. 

the3million Young Europeans have created a document which gives more detail:

These rules are also currently being debated in some countries and they may change. Before applying for British citizenship, do check with your embassy or consulate about your EU country’s rules on this issue.