In 2012, several reports were published advocating the creation of a genuine fiscal capacity at the European level, to strengthen the foundations of the Economic and Monetary Union (EMU) by helping member countries to adjust to asymmetric shocks. The design of a common stabilization fund should be well thought, though: what would be stabilized? When would transfers be paid? Would they be directed towards governments or individuals? How would they be financed? And would they help stabilize economies? We address these questions by outlining recent existing proposals, their features and their simulated stabilization effects. We finally underline the difficulties in implementing such a scheme.