The orientation of the angular momentum vector with respect to the triaxial density distribution selects a left-handed or right-handed system principal axes. This breaking of chiral symmetry manifests itself as pairs of nearly identical DeltaI = 1-bands. The chiral structures combine high-j particles and high-j holes with a triaxial rotor. Tilted axis cranking calculations predict the existence of such configurations in different mass regions. There is experimental evidence in odd-odd nuclei around mass 134. The quantized motion of the angular momentum vector between the left- and right-handed configurations, which causes the splitting between the chiral sister bands, can be classified as tunneling (chiral rotors) or oscillation (chiral vibrators).