In two spatial dimensions quantum mechanical particles are not limited to being bosons or fermions as they are in three dimensions, but can be particles known as anyons. Anyons come in two major flavors — Abelian and non-Abelian — both of which were predicted to occur in the fractional quantum Hall (FQH) states. In this talk I will report on the study of braiding statistics of anyons through detection of phase slips in quantum interference experiments through which the statistical angle of anyon braiding can be measured. In the nu =7/3 FQHE state we confirm the anyonic braiding statistics by detecting the postulated statistical phase angle of 2pi/3. This result is consistent with a change of the anyon number by one. In the interference study of the nu = 5/2 FQH state we observe phase slips of pi/4, pi, and 5pi/4. Phase slips of pi/4 arise from the Abelian braiding of e/4 quasiparticles; the pi phase slips are likely signatures of quasiparticles braiding a neutral fermion, providing compelling support for the existence of non-Abelian anyons.