Note that this is too recent for the rules of this subreddit.

The EU does not enact laws, but so-called directives, which are meant to be transcribed into local laws by local parliaments. There are a number of directives and treaties, such as Schengen, which detail how judicial authorities apply across member states. Typically legal decisions from one country are binding in another.

One example I can think of is that of a French man who unknowingly impregnated a German woman. She went back to Germany, had him acknowledge being the father, and got a local court to order the father to pay child support. Even though child support does not exist in France unless the parents were married at the time of birth or the father signed the birth certificate, he was obliged to pay child support nonetheless.