Employing the quantum interference among one- and two-photon excitations induced by ultrashort two-color laser pulses it is possible to generate charge and spin currents in semiconductors and semiconductor nanostructures on femtosecond time scales. Here, it is reviewed how the excitation process and the dynamics of such photocurrents can be described on the basis of a microscopic many-body theory. Numerical solutions of the semiconductor Bloch equations (SEE) provide a detailed description of the time-dependent material excitations. Applied to the case of photocurrents, numerical solutions of the SEE for a two-band model including many-body correlations on the second-Born Markov level predict an enhanced damping of the spin current relative to that of the charge current. Interesting effects are obtained when the scattering processes are computed beyond the Markovian limit. Whereas the overall decay of the currents is basically correctly described already within the Markov approximation, quantum-kinetic calculations show that memory effects may lead to additional oscillatory signatures in the current transients. When transitions to coupled heavy- and light-hole valence bands are incorporated into the SEE, additional charge and spin currents, which are not described by the two-band model, appear.