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Abstract In this article, we provide notes that complement the lectures on the relativistic Euler equations and shocks that were given by the second author at the programMathematical Perspectives of Gravitation Beyond the Vacuum Regime, which was hosted by the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna in February 2022. We set the stage by introducing a standard first-order formulation of the relativistic Euler equations and providing a brief overview of local well-posedness in Sobolev spaces. Then, using Riemann invariants, we provide the first detailed construction of a localized subset of the maximal globally hyperbolic developments of an open set of initially smooth, shock-forming isentropic solutions in 1D, with a focus on describing the singular boundary and the Cauchy horizon that emerges from the singularity. Next, we provide an overview of the new second-order formulation of the 3Drelativistic Euler equations derived in Disconzi and Speck (2019Ann. Henri Poincare202173–270), its rich geometric and analytic structures, their implications for the mathematical theory of shock waves, and their connection to the setup we use in our 1Danalysis of shocks. We then highlight some key prior results on the study of shock formation and related problems. Furthermore, we provide an overview of how the formulation of the flow derived in Disconzi and Speck (2019Ann. Henri Poincare202173–270) can be used to study shock formation in multiple spatial dimensions. Finally, we discuss various open problems tied to shocks.