We construct the covariant kappa-symmetric superstring action for a type IIB superstring on AdS(5) x S-5 background, The action is defined as a 2d sigma-model on the coset superspace SU(2,2\4)/SO(4,1)xSO(5) and is shown to be the unique one that has the correct bosonic and flat space limits, (C) 1998 Elsevier Science B.V.