In the present paper, we study a new generalization of warped product manifolds, called sequential warped product manifolds, with respect to a semi-symmetric metric connection. We obtain relations for covariant derivatives, Riemannian curvature, Ricci curvature and scalar curvature of the sequential warped product manifolds with respect to the semi-symmetric connection, respectively, and demonstrate the relationship between them and curvatures with respect to the Levi-Civita connection. Also, we consider sequential warped product space-time models, namely sequential generalized Robertson-Walker space-times and sequential standard static space-times, with semi-symmetric metric connections and obtain conditions for such space-times to be Einstein.