Existing theoretical investigations on oscillating foil propulsion have rarely treated a foil which has chordwise flexibility. However, an experimental investigation showed that propulsion performance of a partially elastic oscillating foil Is better than that of a rigid one. In this paper, the authors propose a theory for partially elastic oscillating foils, where the deformation of the elastic part is treated by incorporating the equation of elastic curve in the small amplitude linearized oscillating foil theory. Using this theory, theoretical analysis of propulsion performance were made in detail for both elastic and rigid foils. The following results were obtained. (1) It was shown that propulsion performance of an oscillating foil is decided by three parameters of foil motion, namely, reduced frequency, phase difference between heaving and pitching motion and modified heaving to pitching amplitude ratio. (2) Under the same conditions of motion, compared to a rigid foil, an elastic foil produces less thrust but at a higher efficiency. (3) To obtain the same thrust, an elastic foil is more efficient compared to a rigid one. (4) To produce a given thrust at the highest efficiency, the conditions of motion, namely, heaving and pitching, is not so much affected by the variation in reduced frequency in case of an elastic foil compared to that of a rigid one. This fact is desirable from practical operating conditions.