In Europe, you are by default responsible for copyright infringements and other illegal contents in user posted content.
However, in Europe, the EU Ecommerce directive, Article 14 says this:
1. Where an information society service is provided that consists of the storage of information provided by a recipient of the service, Member States shall ensure that the service provider is not liable for the information stored at the request of a recipient of the service, on condition that:
(a) the provider does not have actual knowledge of illegal activity or information and, as regards claims for damages, is not aware of facts or circumstances from which the illegal activity or information is apparent; or
(b) the provider, upon obtaining such knowledge or awareness, acts expeditiously to remove or to disable access to the information.
While not as clear and explicit as the safe harbor provisions of the DMCA, in practice it means the same:
As an information service provider in Europe, hosting user posted content, you do not become liable for illegal user posted content until you have knowledge about the illegal content (by receiving a formal or informal notice that makes you aware of the existence of the illegal content);
Acts expeditiously to remove or to disable access to the information upon receiving such a notice (e.g. DMCA-style takedown notice). However, while a service provider in the US can safely ignore any notice that does not follow the exact protocol outlined in the DMCA, there are no such formal requirements in Europe.
I.e.: A service provider in Europe will not be liable if they put in their policies that copyright infringements are not allowed and will be removed - and of course: The service provider must also expeditiously enforce this policy.
Europe copyright law does not have the broad exceptions from copyright that is afforded by "fair use" in the USA, but there are some exceptions and limitations to copyright outlined in Article 5 of the EU copyright directive. If user posted content copyrighted by a third party is covered by one of the exceptions and limitations listed in this article, it is not illegal for the user to post it.
As for case law of the Ecommerce directive, see - for example - this analysis of Sweden vs. Pirate Bay.