All European states have a primary obligation to establish and maintain governmental schools everywhere, but as the result of political struggle and constitutional guarantees, they have also allowed and often financed non-state schools based on special pedagogical, religious or philosophical ideas. Depending on the level of state grants for non-state schools, states have more or less the right to supervise these non-governmental schools and seek to guarantee that the quality of organisation and teachers are not lower than those in governmental schools. Using comparable cross-national data for all member states of the European Union, we first describe four existing basic arrangements of non-governmental and governmental schools: integrated educational systems of public and non-state schools, denomination supportive educational systems, limited-support non-governmental schools and educational systems with segregated public and non-state schools. Using the same cross-national data for all member states of the European Union, we then explore three other topics: parental background and the choice of non-governmental schools, non-governmental schools and their cognitive outcomes, and non-governmental schools and their non-cognitive outcomes. There are important differences between non-governmental-"in"dependent (without state grants) and non-governmental-"de"pendent schools (with state grants); that school choice of non-governmental-"de"pendent schools is more related to socially mobile parents, whereas schools choice of non-governmental-"in"dependent schools is more related the reproduction of social classes; that in a majority of European countries, non-governmental-"de"pendent schools are more effective cognitively than governmental schools, but that non-governmental-"in"dependent schools are more effective cognitively only in a few countries and more ineffective in a larger number of countries. Also non-governmental-"de"pendent schools are "not" more effective non-cognitively than governmental schools.