Extensive optical and near-infrared (NIR) observations of the Type IIb supernova (SN IIb) 2008ax are presented, covering the first year after the explosion. The light curve is mostly similar in shape to that of the prototypical SN IIb 1993J, but shows a slightly faster decline rate at late phases and lacks the prominent narrow early-time peak of SN 1993J. From the bolometric light curve and ejecta expansion velocities, we estimate that about 0.07–0.15 M⊙ of 56Ni was produced during the explosion and that the total ejecta mass was between 2 and 5 M⊙, with a kinetic energy of at least 1051 erg. The spectral evolution of SN 2008ax is similar to that of SN Ib/IIb 2007Y, exhibiting high-velocity Ca ii features at early phases and signs of ejecta–wind interaction from Hα observations at late times. NIR spectra show strong He i lines similar to SN Ib 1999ex and a large number of emission features at late times. Particularly interesting are the strong, double-peaked He i lines in late NIR spectra, which – together with the double-peaked [O i] emission in late optical spectra – provide clues for the asymmetry and large-scale Ni mixing in the ejecta.