One of the outstanding open problems in the study of fluids is the global regularity of smooth solutions to the three-dimensional incompressible Euler equation. I will begin by introducing the incompressible Euler equation as well as some classical well-posedness results. Then I will discuss various attempts to understand the global regularity problem and related problems moving into recent results. One popular attempt to understand the global regularity problem is to study simplified lower dimensional models that can be satisfactorily solved. A major issue with studying simplified models is that they may have no bearing on the dynamics of the actual 3d Euler equation--especially since the closer a model gets to modeling the dynamics of 3d Euler the more challenging understanding the dynamics of the model is. In recent works with I. Jeong, we derived a “good” model through the use of symmetry properties of the equation. In particular, we proved that if singularity formation can be established for a particular two-dimensional equation, then there is singularity formation for the full 3d Euler equation for finite-energy solutions lying in a critical space where there is local well-posedness. Similar results can be proven for the surface quasi-geostrophic (SQG) equation arising in atmospheric dynamics and there a one-dimensional model is derived. I will then discuss recent results on some of these models and their implications.