An evaporator with a cool storage function includes a plurality of refrigerant flow tubes. The flow tubes are spaced apart from one another in a thickness direction to form spaces among the plurality of refrigerant flow tubes. Outer fins are disposed in a first part of the spaces and are joined to the plurality of refrigerant flow tubes. The cool storage material container contains a cool storage material and is disposed in a second part of the spaces other than the first part. An inner fin is disposed within the cool storage material container and has crest portions extending along a longitudinal direction of the flow tubes, trough portions extending along the longitudinal direction, and connection portions connecting the crest portions and the trough portions.