The study of hypersurfaces in Euclidean spaces has a long and interesting history.It is obvious that the mean curvature and scalar curvature of a given hypersurface in Euclidean spaces are the basic invariants with important geometric meaning. Now, Let M be a complete two-dimensional surfaces immersed into the three-dimensional Euclidean space. Then a 0, c 0, it is called a sphere space, Euclidean space or hyperbolic space, respectively, and it is denoted by S n+1 (c), R n+1 , or H n+1 (c). It is well known that there are many rigidity results for hypersurfaces with constant mean curvature or with constant scalar curvature in S n+1 (c) or R n+1 , but less are obtioned for hypersurfaces immersed into a hyperbolic space. In this thesis, we consider n-dimensional oriented complete hypersurfaces with constant scalar curvature of a Euclidean space R n+1 . We characterize the hypersurface S k (c) × R n+1 . Now suppose M has non-negative sectional curvature and constant scalar curvature ? c. Then we prove that either M is totally umbilical, the Riemannian product of two totally umbilical constantly curved submanifold, or M is flat. As a corollary, one sees that if the ambient manifold is R n+1 , M must be the sphere and if the ambient manifold is S n+1 , M has the form S n?p × S p . In the course of the proof of this theorem, we introduce some self-adjoint differential operators which are interesting for their own right. When M is complete and non-compact, we assume that M has non-negative curvature and the ambient space is R n+1 . Our conclusion is that M must be a cylinder S p × R n?p . In the present thesis, we obtain new formula (see 3.16) in the case of a hypersurface M immersed with constant mean curvature in a space M of constant sectional curvature and then drive a new formula (see 3.18) for the function f which involves the sectional curvature of M . Based on this new formula our main results are the determination of hypersurfaces M of nonnegative sectional curvature immersed in the Euclidean space R n+1 or the sphere S n+1 with constant mean curvature under the additional assumption that the function f is constant. This additional assumption is automatically satisfied if M is compact. we state the general results in a global form assuming completeness of M . In this thesis, we also investigate the nonnegative sectional curvature hypersurfaces in a real space form M n+1 (c). We obtain some rigidity results of nonnegative sectional curvature hypersurfaces M n+1 (c) with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product S k (a) × S n?k (?1 ? a 2 ), 1 ? k ? n ? 1, in S n+1 (1) and the Riemannian product H k (tanh 2 r ? 1) × S n?k (coth 2 ? 1), 1 ? k ? n ? 1, in H n+1 (?1).