In this paper the concept of passivity is generalized from the viewpoint of the rate of energy dissipation, introducing a new supply rate with an index gamma. A stabilizability result for nonlinear systems is obtained using the argument of the supply rate and the dissipative inequality, In the case of linear systems, it is shown that the index gamma can be used as a measure of phase of transfer functions, which provides a phase curve shaping method, A stability result of a feedback system involving a gamma-positive real system is also obtained and its effectiveness will be shown by an example, Finally, a design method of gamma-passive systems is discussed using a bilinear transformation of nonlinear systems, called Carler-transform.