Electrical and thermal transport properties of quasicrystals are reviewed on the examples of i-Ag–In–Yb and i-Al–Cu–Fe icosahedral phases and d-Al–Co–Ni decagonal phase. Using samples of single-grain morphology and high structural quality, and performing the measurements along well-defined crystallographic directions, the following basic questions in the context of physical properties of quasicrystals are addressed, both experimentally and theoretically: (1) are the unusual transport properties of quasicrystals introduced by the quasiperiodicity of the structure or are they a consequence of complex local atomic order with no direct relationship to the quasiperiodicity; (2) what is the role of the electronic structure of quasicrystals in their electronic transport properties, especially the pseudogap in the electronic density of states in the vicinity of the Fermi energy; (3) what is the anisotropy of the transport coefficients along different crystallographic directions for icosahedral and decagonal quasicrystals and (4) what are the true intrinsic properties of quasicrystalline phases?