I will describe some recent work on the incompressible Euler equations and related partial differential equations specifically related to "Critical Phenomena". It is, by now, known that the incompressible Euler equation is ill-posed in most "critical classes" such as the class of Lipschitz continuous or C^1 velocity fields (even when the data is taken to be smooth away a single point). Despite this, we prove well-posedness (global in 2d and local in 3d) for merely Lipschitz data which is smooth away from the origin and satisfies a mild symmetry assumption. To do this requires a deep understanding of the nature of unboundedness of singular integrals on $L^\infty$. Through this understanding, we define a well-adapted class of critical function spaces in which we prove well-posedness. After this, we extract a simplified equation which is satisfied by "scale invariant" solutions which lie within the setting of our local/global well-posedness theory. These scale-invariant solutions, in the 2d Euler setting, can be shown to have very interesting dynamical properties such as time-quasiperiodic behavior. Moreover, these scale-invariant solutions (while having infinite energy) can be used to prove the existence of finite-energy solutions with the "same" dynamical properties.This is joint work with In-Jee Jeong.