  We consider soft-gluon evolution at the amplitude level. Our evolution
includes Coulomb exchanges and applies to generic hard-scattering processes
involving any number of coloured partons. We emphasise the special role played
by a Lorentz-invariant evolution variable, which coincides with the transverse
momentum of the latest emission in a suitably defined dipole zero-momentum
frame. We also relate the evolution algorithm, which was used originally in the
derivation of super-leading logarithms, to renormalization group evolution
equations that have been encountered recently. Handling large colour matrices
presents the most significant challenge to numerical implementations and we
present a means to expand systematically about the leading colour
approximation.
