  We test several conjectural dualities between strongly coupled superconformal
field theories in three dimensions by computing their exact partition functions
on a three-sphere as a function of Fayet-Iliopoulos and mass parameters. The
calculation is carried out using localization of the path integral and the
matrix model previously derived for superconformal N = 2 gauge theories. We
verify that the partition functions of quiver theories related by mirror
symmetry agree provided the mass parameters and the Fayet-Iliopoulos parameters
are exchanged, as predicted. We carry out a similar calculation for the mirror
of N = 8 super-Yang-Mills theory and show that its partition function agrees
with that of the ABJM theory at unit Chern-Simons level. This provides a
nonperturbative test of the conjectural equivalence of the two theories in the
conformal limit.
