In this talk, I will discuss recent developments in the $L^p$-regularity problem for the Bergman projection on Rudin ball quotients. A Rudin ball quotient is a domain of the form $\varphi^\Gamma(\mathbb{B}_n)$, where $\mathbb{B}_n$ is the unit ball in $\mathbb{C}^n$, $\Gamma$ is a finite unitary reflection group (i.e., a finite subgroup of $U(n)$ generated by reflections), and $\varphi^\Gamma = (\varphi^\Gamma_1,\ldots,\varphi^\Gamma_n): \mathbb{C}^n \to \mathbb{C}^n$ is a standard orbit map associated to $\Gamma$. The components ${\varphi^\Gamma_1, \ldots, \varphi^\Gamma_n}$ form a set of homogeneous basic invariants under the natural action of $\Gamma$ on the ring of holomorphic polynomials on $\mathbb{C}^n$. This talk is based on joint work with Gian Maria Dall’Ara and Debraj Chakrabarti.