  We show that for any transcendental meromorphic function $f$ there is a point
$z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is,
tend to $\infty$, arbitrarily slowly. The proof uses new covering results for
analytic functions. We also introduce several slow escaping sets, in each of
which $f^n(z)$ tends to $\infty$ at a bounded rate, and establish the
connections between these sets and the Julia set of $f$. To do this, we show
that the iterates of $f$ satisfy a strong distortion estimate in all types of
escaping Fatou components except one, which we call a plane-filling wandering
domain. We give examples to show how varied the structures of these slow
escaping sets can be.
