Over the past decade, a number of classical moduli spaces (eg of del Pezzo surfaces, cubic hypersurfaces in P4 and P5, and curves of genus 3 and 4) have been realized as quotients by discrete groups of the complex n-ball or the symmetric domain for O(2,n). This is like the moduli space of genus 1 Riemann surfaces being the quotient of the complex 1-ball (the disk) by SL(2,Z). We will survey results by Kondo, Laza, Looijenga, and others, including work of the speaker and his coauthors.