The Stockholm Programme establishing the political priorities of the area of freedom, security and justice for the period 2010-2014 stated that the EU should seek accession to the 1951 Geneva Convention on the status of refugees and its 1967 Protocol. In spite of the political difficulties that the materialization of this recommendation, repeated in 2011, would encounter, this contribution analyzes the legal implications of a hypothetical accession of the Union to the cornerstone of the international regime on refugee protection, which simultaneously constitute the core of the EU asylum acquis by the direct reference found in art. 78.1 TFEU and art. 18 of the EU Charter of Fundamental Rights. For that purpose, this research firstly concentrates on analyzing the legal feasibility of the accession, both under EU law ⎯focusing on the competence question, looking at its existence and also its nature with regard to Member States’powers⎯, and under international law ⎯examining the terms of the Geneva Convention to identify the need for its revision or adjustment⎯. Secondly, the added value of the accession is assessed, particularly regarding the current and future role and place the Geneva Convention enjoys within the EU legal order.