We propose a construction of the quantum-corrected Coulomb branch of a
general 3d gauge theory with $\mathcal{N}=4$ supersymmetry, in terms of local
coordinates associated with an abelianized theory. In a fixed complex
structure, the holomorphic functions on the Coulomb branch are given by
expectation values of chiral monopole operators. We construct the chiral ring
of such operators, using equivariant integration over BPS moduli spaces. We
also quantize the chiral ring, which corresponds to placing the 3d theory in a
2d Omega background. Then, by unifying all complex structures in a twistor
space, we encode the full hyperkähler metric on the Coulomb branch. We verify
our proposals in a multitude of examples, including SQCD and linear quiver
gauge theories, whose Coulomb branches have alternative descriptions as
solutions to the Bogomolnyi and/or Nahm equations.