The problem of establishing links between the curvature, the algebraic and topological structure of a (pseudo)Riemannian manifold is one of the important problems of the (pseudo)Riemannian geometry. There are several well known researches and results: the Hadamard — Cartan theorem about the complete simply-connected Riemannian manifold of non-positive sectional curvature, Gromov theorem about Riemannian manifolds of nonnegative Ricci curvature, comparison angles triangle theorem by A.D. Alexandrov — V.A. Toponogov, theorem on the sphere, extreme theorem in the Riemannian geometry and some other results. In general, the purpose of the research of (pseudo)Riemannian manifolds with restrictions on curvature of various types is rather complicated. Therefore, it is natural to consider this problem in a narrower class of (pseudo)Riemannian manifolds, for example, in the class of homogeneous (pseudo)Riemannian manifolds and, in particular, in the class of metric Lie groups. The article presents the results of research of (pseudo) Riemannian metrics of sign-definite curvature, signatures of the curvature operators; the questions of the existence of a locally homogeneous (pseudo) Riemannian spaces and, in particular, the metric Lie groups with a given spectrum any operator curvature.