We investigate the consequences of a non-negligible baryon fraction for models of structure formation in cold dark matter dominated cosmologies, emphasizing in particular the existence of oscillations in the present-day matter power spectrum. These oscillations are the remnants of acoustic oscillations in the photon—baryon fluid before the last scattering, for which evidence from measurements of the cosmic microwave background anisotropy is mounting. For acceptable values of the cosmological and baryon densities, the oscillations modulate the power by up to ∼ 10 per cent, with a ‘period’ in spatial wavenumber that is close to δ k ∼ 0.05 Mpc−1. We study the effects of non-linear evolution on these features, and show that they are erased for k≥0.2 hMpc−1. At larger scales, the features evolve as expected from second-order perturbation theory: the visibility of the oscillations is affected only weakly by non-linear evolution. No realistic CDM parameter combination is able to account for the claimed feature at k≈0.1 h−1Mpc in the APM power spectrum, or the excess power at 100 h−1Mpc wavelengths quoted by several recent surveys. Thus baryonic oscillations are not predicted to dominate existing measurements of clustering. We examine several effects that may mask the features that are predicted, and conclude that future galaxy surveys may be able to detect the oscillatory features in the power spectrum provided baryons comprise ≥15 per cent of the total density, but it will be a technically challenging achievement.