A (p_q, n_k) configuration is a collection of p points and n lines in which every point has q lines incident with it and every line has k points incident with it. The lines may be purely combinatorial, viewed simply as collections of points; they may be topological, so that any pair intersects transversally exactly once but the lines themselves are allowed to wiggle around; or they may be geometric, as actual straight lines. This talk will provide background and discussion about a number of open problems in the study of combinatorial, topological, and geometric configurations, as well as discussing some recent results and relations to other areas of discrete geometry.