  We study the iteration of functions in the exponential family. We construct a
number of sets, consisting of points which escape to infinity `slowly', and
which have Hausdorff dimension equal to 1. We prove these results by using the
idea of an annular itinerary. In the case of a general transcendental entire
function we show that one of these sets, the uniformly slowly escaping set, has
strong dynamical properties and we give a necessary and sufficient condition
for this set to be non-empty.
