Article 55 answers a foundational enforcement question: which supervisory authority is legally entitled to deal with a particular processing operation? Its three paragraphs establish the general territorial rule, a special rule for public-authority and public-interest processing, and an exclusion protecting courts when they act judicially.
The Article is best understood as a rule about regulatory competence, not merely geography. It determines which authority may receive complaints, investigate facts, demand information, make findings, issue orders and impose sanctions. It must be read together with Articles 3, 4(23), 51, 56, 60 to 62 and 77 GDPR. The official text and recitals show that Article 55 creates the ordinary national competence rule, while Article 56 reallocates responsibility in qualifying cross-border cases.europa+1