  The anyonic quantum walk is a dynamical model describing a single anyon
propagating along a chain of stationary anyons and interacting via mutual
braiding statistics. We review the recent results on the effects of braiding
statistics in anyonic quantum walks in quasi-one dimensional ladder geometries.
For anyons which correspond to spin-1/2 irreps of the quantum groups $SU(2)_k$,
the non-Abelian species $(1<k<\infty)$ gives rise to entanglement between the
walker and topological degrees of freedom which is quantified by quantum link
invariants over the trajectories of the walk. The decoherence is strong enough
to reduce the walk on the infinite ladder to classical like behaviour. We also
present numerical results on mixing times of $SU(2)_2$ or Ising model anyon
walks on cyclic graphs. Finally, the possible experimental simulation of the
anyonic quantum walk in Fractional Quantum Hall systems is discussed.
