The motion of the ideal incompressible fluid is described by the Euler equations. Their solution $u(x,t)$ exists for any initial velocity field $u_0$ provided it is regular enough. The solution has the same regularity as the initial velocity $u_0$. However, all the particle trajectories are analytic curves! This striking fact was proved in 2013 (Frisch&Zheligowsky, Nadirashvili, Shnirelman), while it could be proved back in 1925 by Lichtenstein who had all the necessary ideas. In fact, this result is a consequence of an analytic structure on the group of volume preserving diffeomorphisms.   Other related subject is the structure of complex singularities of real-analytic solutions of the Euler equations. Using appropriate functional spaces, we are able to construct simple complex singularities of stationary and non-stationary solutions.