In this paper we summarize recent diffusion studies from our laboratory. Diffusion of the following elements has been studied in icosahedral quasicrystals: Mn, Fe, Ga, In and Zn in AlPdMn and Zn in ZnMgRE (RE=rare earth metal). The diffusivities follow Arrhenius-like temperature dependencies. In addition, for Mn and Zn in icosahedral AlPdMn we have studied the dependence of the diffusivities on hydrostatic pressure. Signs and values of the activation volumes around 0.7 atomic volumes are strongly in favour of a vacancy-type diffusion mechanism. Diffusion of Co and Ni has been studied in both principal directions of decagonal AlNiCo quasicrystals. The small diffusion anisotropy is a strong argument in favour of vacancy-mediated diffusion. Significant contributions of phason flips to diffusion should result in faster diffusion within the layers perpendicular to the decagonal axis, which is not observed.