  Quantum ladder models, consisting of coupled chains, form intriguing systems
bridging one and two dimensions and have been well studied in the context of
quantum magnets and fermionic systems. Here we consider ladder systems made of
more exotic quantum mechanical degrees of freedom, so-called non-Abelian
anyons, which can be thought of as certain quantum deformations of ordinary
SU(2) spins. Such non-Abelian anyons occur as quasiparticle excitations in
topological quantum fluids, including p_x + i p_y superconductors, certain
fractional quantum Hall states, and rotating Bose-Einstein condensates. Here we
use a combination of exact diagonalization and conformal field theory to
determine the phase diagrams of ladders with up to four chains. We discuss how
phenomena familiar from ordinary SU(2) spin ladders are generalized in their
anyonic counterparts, such as gapless and gapped phases, odd/even effects with
the ladder width, and elementary `magnon' excitations. Other features are
entirely due to the topological nature of the anyonic degrees of freedom.
