This talk is concerned with the properties of the escaping set of a
transcendental meromorphic function (that is, the set of points that tend to
infinity under iteration). We discuss the relationship between the escaping
set and the Fatou set (the stable set) and the Julia set (the chaotic set).
We also discuss the different rates at which points can escape to infinity
and show how the notion of the fast escaping set enables us to make progress
on a conjecture of Eremenko that all the components of the escaping set are
unbounded. We will illustrate all of these ideas with various examples,
including examples of functions for which the escaping set is in fact connected.