  We develop the general quantum measurement theory of non-Abelian anyons
through interference experiments. The paper starts with a terse introduction to
the theory of anyon models, focusing on the basic formalism necessary to apply
standard quantum measurement theory to such systems. This is then applied to
give a detailed analysis of anyonic charge measurements using a Mach-Zehnder
interferometer for arbitrary anyon models. We find that, as anyonic probes are
sent through the legs of the interferometer, superpositions of the total
anyonic charge located in the target region collapse when they are
distinguishable via monodromy with the probe anyons, which also determines the
rate of collapse. We give estimates on the number of probes needed to obtain a
desired confidence level for the measurement outcome distinguishing between
charges, and explicitly work out a number of examples for some significant
anyon models. We apply the same techniques to describe interferometry
measurements in a double point-contact interferometer realized in fractional
quantum Hall systems. To lowest order in tunneling, these results essentially
match those from the Mach-Zehnder interferometer, but we also provide the
corrections due to processes involving multiple tunnelings. Finally, we give
explicit predictions describing state measurements for experiments in the
Abelian hierarchy states, the non-Abelian Moore-Read state at $\nu=5/2$ and
Read-Rezayi state at $\nu = 12/5$.
