  This paper addresses a long standing problem - to identify the chiral ring
and moduli space (i.e. as an algebraic variety) on the Coulomb branch of an N =
4 superconformal field theory in 2+1 dimensions. Previous techniques involved a
computation of the metric on the moduli space and/or mirror symmetry. These
methods are limited to sufficiently small moduli spaces, with enough symmetry,
or to Higgs branches of sufficiently small gauge theories. We introduce a
simple formula for the Hilbert series of the Coulomb branch, which applies to
any good or ugly three-dimensional N = 4 gauge theory. The formula counts
monopole operators which are dressed by classical operators, the Casimir
invariants of the residual gauge group that is left unbroken by the magnetic
flux. We apply our formula to several classes of gauge theories. Along the way
we make various tests of mirror symmetry, successfully comparing the Hilbert
series of the Coulomb branch with the Hilbert series of the Higgs branch of the
mirror theory.
