  Quantum mechanical systems, whose degrees of freedom are so-called su(2)_k
anyons, form a bridge between ordinary SU(2) spin systems and systems of
interacting non-Abelian anyons. Such a connection can be made for arbitrary
spin-S systems, and we explicitly discuss spin-1/2 and spin-1 systems. Anyonic
spin-1/2 chains exhibit a topological protection mechanism that stabilizes
their gapless ground states and which vanishes only in the limit (k to
infinity) of the ordinary spin-1/2 Heisenberg chain. For anyonic spin-1 chains
we find their phase diagrams to closely mirror the one of the biquadratic SU(2)
spin-1 chain. Our results describe at the same time nucleation of different 2D
topological quantum fluids within a `parent' non-Abelian quantum Hall state,
arising from a macroscopic occupation of localized, interacting anyons. The
edge states between the `nucleated' and the `parent' liquids are neutral, and
correspond precisely to the gapless modes of the anyonic chains.
