We describe an ab initio and nonperturbative 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 it generalizes previous time-dependent R -matrix techniques restricted to linearly polarized fields. We discuss both the fundamental equations, required to propagate the multielectron wave function 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 2 p electrons is quantified.