There is more than one way of finding research funding at European universities. Depending on your status (e.g. non-EU researcher, young researcher etc.), you can get research funding from various sources.
In any case, you don’t necessarily have to be a citizen or a resident of a particularly country in order to be funded by national scholarships and grants.
EU research funding
The European Union offers various opportunities for researchers to access financial support. Some of the main sources of EU research funding are:
The Intra-European Fellowships scheme provides funding for 12-24 months, for researchers who want to advance their career in Europe outside their home country. Funding covers advanced training needs, such as new and/or complementary skills, and transnational mobility. Applications are open to researchers based in EU Member States or Associated Countries.
International Incoming Fellowships provide support for researchers that move from non-EU countries to EU Member States or Associated Countries, again for 12-24 months. IIF applications should be made in liaison with the organization or institution that would be willing to host you in Europe (including universities, research centers, companies).
The International Research Staff Exchange Scheme (IRSES) finances active research organizations (e.g. universities, research centers etc.), for a period of 24-48 months. The allowance covers mobility costs (including travel costs) for staff members from an EU Member State or Associated Country staying in an eligible non-EU country. It may also cover the costs of stays by staff from certain non-EU countries in Europe.