"We study circle patterns that locally correspond to weighted Delaunay triangulations of configurations of disjoint disks. These patterns, when modeled on metric geometries such as the Euclidean or hyperbolic plane, have been successfully applied to the study of discrete conformal maps. They exhibit discrete uniformization theorems and possess strong approximation properties. Moreover, when considered at the ideal boundary of hyperbolic 3-space, these discrete conformal mapping problems relate to the realization of hyperbolic surfaces as boundaries of periodic convex polyhedra. The concept of a circular disk — and by extension, hyperideal circle patterns — naturally extends to the broader setting of the complex projective plane. In this talk, I will introduce the Möbius-theoretic framework required to describe the configuration spaces of hyperideal circle patterns and discuss their associated realization problems."