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 aMach-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 v = 5/2 and Read-Rezayi state at v = 12/5.