Quasicrystals were discovered in the early 1980s in metallic alloys. Their diffraction pattern consists of Bragg peaks, like in a periodic crystal, but with a crystallographically forbidden symmetry like icosahedral symmetry which has an axis of fivefold rotational symmetry. Other quasicrystals are periodic along one axis, but feature twelve-, ten-, or eightfold rotational symmetry in the plane around this axis. These dodecagonal, decagonal, and octagonal quasicrystals show anisotropic physical properties. – The diffraction patterns can be resolved by interpreting quasicrystals as aperiodically ordered solids. Their structure is usually described in terms of quasiperiodic tilings of space, which play the role of the lattice structure in conventional crystals. These tilings can be derived as projections of sections from higher-dimensional periodic lattices, which can accommodate the symmetries that are observed in quasicrystals. – Beside the Bragg peaks, further evidence for the quasiperiodic structure is obtained by high-resolution electron microscopy and atomic force microscopy. The observed surface structures are usually in good correspondence with perfect quasiperiodic tilings, which confirms that the surfaces are consistent terminations of the bulk structure and no significant reconstruction occurs. – Although quasicrystals are usually composed of metallic elements, they show very low electric conductivity which decreases with temperature and decreases also with the structural perfection of the quasicrystal. Other transport properties are also interesting, such large thermoelectric powers and Hall coefficients that depend crucially on the composition and may even undergo sign changes when the temperature changes. – The unusual physical properties of these quasicrystals as well as mathematical models and aspects of material science and technical applications will be discussed in this lecture. In addition, the appearance of quasiperiodic features in nature, art, and architecture will be considered.