We present a comprehensive study of Bi thin-film growth on quasicrystal surfaces. The substrates used for the growth are the fivefold surface of icosahedral (i)-Al-Cu-Fe and i-Al-Pd-Mn and the tenfold surface of decagonal (d)-Al-Ni-Co quasicrystals. The growth is investigated at 300 and 525 K substrate temperatures and at different coverage (θ) ranging from submonolayer to ten monolayers. The film is characterized by scanning tunneling microscopy, reflection high-energy electron diffraction, and x-ray photoelectron spectroscopy. At 300 K, the deposited Bi yields a quasicrystalline film for θ≤1. For 1<θ<5, it forms nanocrystallites with (100) surface orientation. The islands have magic heights, which correspond to the stacking of four atomic layers (predominantly). The selection of magic heights is interpreted in terms of quantum size effects arising from the electron confinement within the film thickness. The islands establish rotational epitaxial relationship with the substrate. For higher coverage, the film grows with monatomic height, not with magic heights, and reflects the symmetry of the bulk Bi. When deposition is performed at 525 K, terrace diffusion is more effective, resulting in the aggregation of Bi adatoms developing into a smooth monolayer with quasiperiodic order. At this temperature, multilayers do not adsorb.