  We describe an ab initio and non-perturbative $R$-matrix with time-dependence
theory for ultrafast atomic processes in light fields of arbitrary
polarization. The theory is applicable to complex, multielectron atoms and
atomic ions subject to ultrashort (particularly few-femtosecond and attosecond)
laser pulses with any given ellipticity, and generalizes previous
time-dependent $R$-matrix techniques restricted to linearly polarized fields.
We discuss both the fundamental equations, required to propagate the
multielectron wavefunction in time, as well as the computational developments
necessary for their efficient numerical solution. To verify the accuracy of our
approach, we investigate the two-photon ionization of He, irradiated by a pair
of time-delayed, circularly polarized, femtosecond laser pulses, and compare
photoelectron momentum distributions, in the polarization plane, with those
obtained from recent time-dependent close-coupling calculations. The predictive
capabilities of our approach are further demonstrated through a study of
single-photon detachment from F$^{-}$ in a circularly polarized, femtosecond
laser pulse, where the relative contribution of the co- and counter-rotating
$2p$ electrons is quantified.
