The ECJ's searchable website, CURIA, includes a list of recent ECJ judgments/Advocate General opinions and a list of recent General Court judgments. These lists link to texts. TIP: Click advanced search.
HUDOC is the database of the European Court of Human Rights. According to The court in brief brochure, the European Court of Human Rights "rules on individual or State applications alleging violations of the civil and political rights set out in the European Convention on Human Rights." The European Convention on Human Rights is a treaty that applies to Council of Europe members. The Council of Europe's website states: "No country has ever joined the EU without first belonging to the Council of Europe."
See this Court's Note explaining the mode of citation and how to refer to the judgments and decisions of the Court (old and new).