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I will give a pedagogical introduction to the incipient mathematical theory of globally hyperbolic supersymmetric configurations in four-dimensional Lorentzian supergravity. First, I will introduce the basics of supergravity in four dimensions as well as the notion of globally supersymmetric configuration as a solution to a system of first-order spinorial differential equations, that is, the supergravity spinorial equations. Then, I will introduce the theory of spinorial polyforms associated with bundles of irreducible real Clifford modules, which provides a convenient geometric framework to study the global geometric and topological properties of the solutions to the supergravity spinorial equations. Then, I will consider the evolution problem for globally hyperbolic supersymmetric configurations, focusing on the constraint equations, their moduli of solutions, and the construction of explicit evolution flows, which we reformulate as the supergravity flow equations for a coupled family of functions and global co-frames on a Cauchy hypersurface. This will lead us to explore in detail the case of (possibly Einstein) globally hyperbolic Lorentzian four-manifolds equipped with a parallel or Killing spinor, obtaining several results about the differentiable topology and geometry of such manifolds. Finally, I will mention several open problems and open directions for future research.