Anyonic Dirac matter is a hypothetical field which is rather unexplored to date. An anyon is a type of quasiparticle that can only occur in two-dimensional systems. Considering bosons and fermions, the interchange of two particles contributes a factor of 1 or -1 to the wave function. In contrast, the operation of exchanging two identical anyons causes a global phase shift. Anyons are generally classified as abelian or non-abelian, according to whether the elementary excitations of the theory transform under an abelian representation of the braid group or a non-abelian one. Abelian anyons have been detected in connection to the fractional quantum Hall effect. The possible construction of anyonic Dirac matter relies on the symmetry protection of crossings of anyonic energy bands. In comparison to bosons and fermions the situation gets more complicated as translations in space do not necessarily commute. Additionally, for given spatial symmetries, the group structure describing the anyon strongly depends on the specific phase of the anyon interchange. For example, for bosons, a rotation of a particle about 2 π i.e., 360 ∘ ∘ , will not change its wave function. For fermions, a rotation of a particle about 2 π, will contribute a factor of − − 1 to its wave function, whereas a 4 π rotation, i.e., a rotation about 720 ∘ ∘ , will give the same wave function as before. For anyons, an even higher degree of rotation can be necessary, e.g., 6 π, 8 π, etc., to leave the wave function invariant.