Let
f
f
be a transcendental entire function and
U
U
be a Fatou component of
f
f
. We show that if
U
U
is an escaping wandering domain of
f
f
, then most boundary points of
U
U
(in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of
U
U
are escaping, then
U
U
is an escaping Fatou component. Some applications of these results are given; for example, if
I
(
f
)
I(f)
is the escaping set of
f
f
, then
I
(
f
)
∪
{
∞
}
I(f)\cup \{\infty \}
is connected.