The generation and propagation of harmonics in an atomic gas are described for the case of an incident Bessel-Gauss beam. Theoretical expressions are derived for the far-field amplitude of the harmonic field by solving the propagation equation using an elaborate integral formalism. We establish simple rules which determine the optimum Bessel-Gauss beam with respect to phase-matching as a function of the medium properties, such as the dispersion and the gas density. Target depletion due to photoionization and refractive index variations originating from both free electrons and dressed linear atomic susceptibilities are taken into account. The intensity-dependent complex atomic dipole moment is calculated using nonpertur- bative methods. Numerical propagation calculations for hydrogen, xenon and argon are presented. For hydrogen we consider the third harmonic of a 355-nm, 15-ps pump beam up to 3 X 10(^13) W/cm(^2) intensity, similarly for xenon, but at lower intensities. For argon we consider the 17th and 19th harmonic of a 810-nm, 30-fs pump beam around 10(^14) W/cm(^2) intensity. We compare conversion efficiencies and both spatial and temporal far-field profiles for an optimized Bessel-Gauss beam with respect to a Gaussian beam of same power and/or peak focal intensity. For the case of hydrogen, we investigate the effect of an ac-Stark-shift induced atomic resonance. We find all results in good agreement with our theoretical predictions. We conclude from our studies that Bessel-Gauss beams can perform better in terms of conversion efficiency than a comparable Gaussian beam. We find this to originate essentially from the more flexible phase-matching conditions for Bessel-Gauss beams. Bessel-Gauss beams also allow for spatial separation of the harmonic and the incident field in the far-field region, owing to the conical shape of their spatial far-field profile. Both features make Bessel-Gauss beams an attractive alternative to Gaussian beams in a limited but substantial number of experimental conditions.