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The 3D incompressible Euler equations are the fundamental model of inviscid incompressible fluids. The issue of regularity of solutions to the Euler equations and its relation to the emergence of turbulence remains a major open problem of fluid dynamics. In this talk, we will explore these concepts from the viewpoint of recent analytical results which are concerned with instantaneous loss of Sobolev regularity of unique local-in-time solutions. We will discuss possible mechanisms of growth of solutions of the 2D Euler equations, as well as the surface quasi-geostrophic equation (SQG), which is closely related to the 3D Euler equations. We will demonstrate that there exist unique solutions to the 2D Euler equations which admit a dramatic instantaneous loss of regularity; namely a gap loss of the order of Sobolev regularity. We will also show how the analysis differs in the case of the SQG equation, and discuss a result which gives examples of solutions losing Sobolev regularity continuously in time. Both results are the first results of these kinds in mathematical fluid mechanics, and develop new analytical tools for studying phenomenology of inviscid incompressible fluids.