  We investigate the effects of quenched randomness on topological quantum
phase transitions in strongly interacting two-dimensional systems. We focus
first on transitions driven by the condensation of a subset of fractionalized
quasiparticles (`anyons') identified with `electric charge' excitations of a
phase with intrinsic topological order. All other anyons have nontrivial mutual
statistics with the condensed subset and hence become confined at the anyon
condensation transition. Using a combination of microscopically exact duality
transformations and asymptotically exact real-space renormalization group
techniques applied to these two-dimensional disordered gauge theories, we argue
that the resulting critical scaling behavior is `superuniversal' across a wide
range of such condensation transitions, and is controlled by the same
infinite-randomness fixed point as that of the 2D random transverse-field Ising
model. We validate this claim using large-scale quantum Monte Carlo simulations
that allow us to extract zero-temperature critical exponents and correlation
functions in (2+1)D disordered interacting systems. We discuss generalizations
of these results to a large class of ground-state and excited-state topological
transitions in systems with intrinsic topological order as well as those where
topological order is either protected or enriched by global symmetries. When
the underlying topological order and the symmetry group are Abelian, our
results provide prototypes for topological phase transitions between distinct
many-body localized phases.
