  In this work we construct Gaussian beam approximations to solutions of the
high frequency Helmholtz equation with a localized source. Under the assumption
of non-trapping rays we show error estimates between the exact outgoing
solution and Gaussian beams in terms of the wave number $k$, both for single
beams and superposition of beams. The main result is that the relative local
$L^2$ error in the beam approximations decay as {$k^{-N/2}$ independent of
dimension and presence of caustics, for $N$-th order beams.
