EURid is the registry operator for the .eu top-level domain. It is a Belgian non-profit association (vzw) with its office in Diegem, Belgium, and it works through accredited registrars that sell .eu names to customers. Registrants manage their names through their registrar account and EURid's own account and WHOIS tools. The service is delivered as an online registry, not as software that organisations install.
As a non-profit association, EURid has no commercial shareholders, so no non-European parent company applies. Its GDPR statement says that it acts as controller for domain holders' registration data and that personal data is stored on servers located in EU countries. It also reports a data protection officer and ISO/IEC 27001 and ISO 22301 certifications. For European organisations, the .eu registry is run within the EU, although the server locations are not named.