The combinatorics of open covers is a study of Cantor’s diagonal argument in various contexts. The field has its roots in a few basic selection principles that arose from the study of problems in analysis, dimension theory, topology and set theory. The reader will also find that some familiar works are appearing in new clothes in our survey. This is particularly the case in connection with such problems as determining the structure of compact scattered spaces and a number of classical problems in topology. We hope that the new perspective in which some of these classical enterprises are presented will lead to further progress. In this article we also attempt to give the reader an overview of the problems and techniques that are currently fueling much of the rapidly increasing current activity in the combinatorics of open covers.