Positional game theory studies combinatorial games of complete information. It is perhaps easiest to think of positional games as generalizations of tic-tac-toe where the game board is an arbitrary hypergraph. (The vertices of the hypergraph are the "positions" that the two players occupy, and the edges of the hypergraph are the "winning sets.") In this talk we will give a brief introduction to positional game theory, including the pivotal theorem of Erdős and Selfridge, and we will discuss some of the extremal systems for that theorem.