  We provide accurate expressions for the $s$-wave scattering length for a
Gaussian potential well in one, two and three spatial dimensions. The Gaussian
potential is widely used as a pseudopotential in the theoretical description of
ultracold atomic gases, where the $s$-wave scattering length is a physically
relevant parameter. We first describe a numerical procedure to compute the
value of the $s$-wave scattering length from the parameters of the Gaussian but
find that its accuracy is limited in the vicinity of singularities that result
from the formation of new bound states. We then derive simple analytical
expressions that capture the correct asymptotic behavior of the $s$-wave
scattering length near the bound states. Expressions that are increasingly
accurate in wide parameter regimes are found by a hierarchy of approximations
that capture an increasing number of bound states. The small number of
numerical coefficients that enter these expressions is determined from accurate
numerical calculations. The approximate formulas combine the advantages of the
numerical and approximate expressions, yielding an accurate and simple
description from the weakly to the strongly interacting limit.
