In this talk, we will study the hydrostatic Euler equations, which describe the leading order behavior of an ideal flow moving in a very narrow domain. After a brief survey of the current progress of the hydrostatic Euler equations, we will establish the local existence and uniqueness of H^s solutions under the local Rayleigh condition. In addition, we will also prove the weak-strong uniqueness, the mathematical justification of the formal derivation and the stability of the hydrostatic Euler equations. These results are based on weighted H^s estimates, which come from a new type of nonlinear cancellations.