In inflationary scenarios with more than one scalar field, inflation may proceed even if each of the individual fields has a potential too steep for that field to sustain inflation on its own. We show that scalar fields with exponential potentials evolve so as to act cooperatively to assist inflation, by finding solutions in which the energy densities of the different scalar fields evolve in fixed proportion. Such scaling solutions exist for an arbitrary number of scalar fields, with different slopes for the exponential potentials, and we show that these solutions are the unique late-time attractors for the evolution. We determine the density perturbation spectrum produced by such a period of inflation, and show that with multiple scalar fields the spectrum is closer to the scale-invariant than the spectrum that any of the fields would generate individually.