The famous Koebe-Andreev-Thurston Theorem guarantees the existence and essential uniqueness of circle packings filling the unit disk. In a series of papers Oded Schramm and Zheng-Xu He proved very general results on the existence of structures formed by ``packable sets'' filling Jordan domains. Though He and Schramm also state some uniqueness results, they did not investigate this question to its full extent - which had a good reason.

The talk is devoted to finite circle agglomerations (which comprise circle packings) filling arbitrary bounded simply connected domains without regularity assumptions on the boundary. In the first part we introduce an appropriate setting, which requires the concept of prime ends, present some general existence results and sketch an elementary proof based on Sperner's lemma.

In the second part we address the issue of uniqueness in some detail. The three relevant criteria concern: the structure of the contact graph, special propertiesof the domain, and the type of ``normalization conditions'' intended to eliminate conformal automorphisms of the domain. Sufficient uniqueness conditions are given for a discrete equivalent of Carathéodory's normalization, prescribing the images of three points (prime ends) on the boundary.

Discussing other types of side conditions, we make it plausible why the naive idea, to fix the center of an interior circle and the direction to the center of a neighbor, is inappropriate to guarantee uniqueness, even for smooth domains.

In the end we propose a modification of the naive standard normalization and formulate a conjecture about existence and uniqueness of non-degenerate circle packings filling arbitrary domains. Its confirmation would yield a faithful discrete counterpart to Riemann's mapping theorem.

The results presented in the talk are based on joint work with David Krieg.