The topic was started in the last quarter of the nineteenth century, and enjoyed a significant popularity until the early twentieth century. Besides enumerations of possible configurations of certain small sizes, the two main results were:

1. A method of inductive construction of all combinatorial configurations (n)—that is, those in which each line is incident with three points, and each point with three lines (Martinetti 1887).
2. A method of geometric realization (by points and straight lines) of all combinatorial configurations (n) if one of the lines is deleted (or, equivalently, admitted as possibly a circle) (Steinitz 1894).

Here is a short description of the topic of configurations for those who are new to it. A configuration (n_k) is a collection of n points and a collection of n lines, each incident with k of the other kind. In a geometric sense "incidence" can be interpreted as "line" containing "point"; in combinatorial interpretation "lines" are sets consisting of "points" incident with them. Most earlier investigations dealt with (n) configurations; the study of (n₄) configurations took off only during recent years, and is being vigorously pursued on both sides of the Atlantic.

Like many other simple concepts about elementary-geometric entities, configuration of points and lines are related in intriguing and nontrivial ways to many branches of mathematics. In particular, these ramifications go from pure combinatorics and graph theory to algebraic geometry of cubic curves, from symmetry and group theory to problems about trigonometric polynomials, from Diophantine equations to questions of computer implementation of construction and enumeration. I expect to give at least a taste of most of these aspects, with information about the currently known facts (published and unpublished). A particularly intriguing collection of facts deals with the interplay of geometric continuity with combinatorial discreteness.