Abstract: In this talk, I will review recent results about existence and qualitative properties of solutions to critical Dirac equations on the Euclidean space. They arise in various problems, both from mathematical physics, namely in the study of nonlinear wave propagation in two-dimensional structures, and differential geometry, for instance in the study of the spinorial version of the Yamabe problem. The emphasis will be on general features of the problem and on key ideas of the proofs. The result presented have been obtained in a series of works, in collaboration with R.L. Frank, A. Maalaoui, A. Malchiodi, V. Martino and R. Wu.