The Bergman projection is a fundamental operator in complex analysis in one and several complex variables. Consequently, its regularity properties on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results in this direction on strongly pseudoconvex domains with near minimal boundary smoothness. In particular, weighted L^p estimates are obtained on strongly pseudoconvex domains of class C^4, where the weight belongs to a suitable generalization of the Bekolle-Bonami class. For such domains, precise estimates on the Bergman kernel function are unavailable. Consequently, we use a kernel free, operator-theoretic technique that goes back to Kerzman, Stein, and Ligocka, and was subsequently refined by Lanzani and Stein to prove (unweighted) L^p regularity. We will also discuss the fundamental obstruction to improving this result to domains of class C^2 (minimal smoothness). This talk is based on joint work with Brett Wick.