The enhanced star-forming activity, typical of starburst galaxies, powers strong galactic winds expanding on kiloparsec (kpc) scales and characterized by bubble structures. Here we discuss the possibility that particle acceleration may take place at the termination shock of such winds. We calculate the spectrum of accelerated particles and their maximum energy, which turns out to range up to a few hundred petaelectronvolt (PeV) for typical values of the parameters. Cosmic rays accelerated at the termination shock are advected towards the edge of the bubble excavated by the wind and eventually escape into extragalactic space. We also calculate the flux of gamma-rays and neutrinos produced by hadronic interactions in the bubble, as well as the diffuse flux resulting from the superposition of the contribution of starburst galaxies on cosmological scales. Finally, we compute the diffuse flux of cosmic rays from starburst bubbles and compare it with existing data.