Harris's inequality and its descendants

In 1960 T.E.Harris proved a fundamental early result about percolation probabilities. As part of his proof he needed an inequality which he described in the following words: "Its truth seems obvious but ...."

In this talk I shall survey some of the classical correlation inequalities which generalise Harris's inequality and were proved in the period 1960 - 1980.

I will also discuss some applications and more recent open problems in the area of combinatorial probability.