In this paper, we study the Ricci soliton in the Riemannian products$M=R^n \times B$ and warped products $M=R \times _f B$ of theEuclidean space and Riemannian manifolds, and the gradient Riccisoliton in the warped products $M=S^1 \times _f B$ of 1-dimensionalcircle and Riemannian manifolds. Moreover, we introduce the concept ofthe generalized Ricci soliton and we suggest the method of constructionof the Riemannian manifold $(M, g)$ with a Ricci soliton $g$.Finally, we also study the Lorentzian warped products with the Riccisoliton.