in this talk, i will discuss characterizations of stationary solutions of the 2d euler equations in two different directions under different assumptions: rigidity (is every stationary solution radial?) and flexibility (do there exist non-radial stationary solutions?). the proofs will have a calculus of variations' flavor, a new observation that finite energy, stationary solutions with simply-connected vorticity have compactly supported velocity, and an application of the nash-moser iteration procedure. based on joint work with jaemin park, jia shi and yao yao.