Consider the 3-dimensional $N=4$ supersymmetric gauge theory associated with a compact Lie group K and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an $SU(2)$-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with symplectic form on its regular locus. It is defined as the spectrum of the equivariant BM homology of a certain moduli space with the convolution product.