Nowadays, contemporary European states are undergoing dynamic transformations connected with the integration processes. Thus, national basic laws not only register to a wider extent, but also become open to integration processes, which is commonly called “Europeanization of constitutions”. Regional integration causes far-reaching constitutional consequences for the European states. It poses new challenges to the EU Member States in the process of adjusting their legal systems, including the contents of their basic laws, to the permanently changing political and legal reality. Objectives and functions performed by the contemporary European states, particularly their scope and ways of fulfilment, are also undergoing transformation. The key issue is also new understanding of the concept of sovereignty.