A refrigerator utilizes in some instances an icemaker having a tandem arrangement of storage bins. A first storage bin receives ice produced by an icemaker and includes a reversible ice mover that when operated in a first direction feeds an ice dispenser, and when operated in a second direction feeds a second storage bin disposed below the first storage bin. A refrigerator may also in another aspect include an externally-accessible ice dispenser having a dispenser recess portion that is removable from the refrigerator when the doors of the refrigerator are closed, e.g., for the purpose of accessing an ice storage bin.