We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of
s
l
2
\mathfrak {sl}_2
and of Heisenberg Lie algebras, Smith’s generalizations of
U
(
s
l
2
)
U(\mathfrak {sl}_2)
, various quantum analogues of these algebras, and many others. We show that the Whittaker modules
V
=
A
w
V = Aw
of the generalized Weyl algebra
A
=
R
(
ϕ
,
t
)
A = R(\phi ,t)
are in bijection with the
ϕ
\phi
-stable left ideals of
R
R
. We determine the annihilator
Ann
A
(
w
)
\operatorname {Ann}_A(w)
of the cyclic generator
w
w
of
V
V
. We also describe the annihilator ideal
Ann
A
(
V
)
\operatorname {Ann}_A(V)
under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant’s well-known results on Whittaker modules and their associated annihilators for
U
(
s
l
2
)
U(\mathfrak {sl}_2)
.