We present plane-wave-type solutions to the superstring effective action which have unbroken space-time supersymmetries. They describe dilaton, axion and gauge fields in a generalization of the Brinkmann metric. A crucial property of the solutions is a conspiracy between the metric and the axion field. Furthermore, due to a relation between the geometry and the gauge fields, the ot' string corrections to the effective on-shell action and to the solutions themselves vanish. We call these solutions supersymmetric string waves.