In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds which later are called a Kenmotsu manifold. In this paper, we study Kenmotsu manifolds with (2n + s)-dimensional s-contact metric manifold that we call generalized Kenmotsu manifolds. Necessary and sufficient condition is given for a s-contact metric manifold to be a generalized Kenmotsu manifold. We show that a generalized Kenmotsu manifold is a locally warped product space. In addition, we study some curvature properties of generalized Kenmotsu manifolds. Moreover, we obtain that the phi-sectional curvature of any semi-symmetric and projective semi-symmetric (2n + s)-dimensional generalized Kenmotsu manifold is -s.