The density-matrix renormalization-group algorithm is extended to treat time-dependent problems. The method provides a systematic and robust tool to explore out-of-equilibrium phenomena in quantum many-body systems. We illustrate the method by showing that attractive interactions enhance the tunneling current between two Luttinger liquids, whereas repulsive interactions suppress it, in qualitative agreement with analytical predictions. Enhancement of the transport current through a quantum dot in the Kondo regime is also exhibited.