Analogously to a notion of curvature of a curve and a surface, in the differential geometry, in the main part of this paper the notion of curvature of hyper-dimensional vector spaces of \textit{Riemannian} metric is generally defined. The defined notion of curvature of \textit{Riemannian} spaces of higher dimensions $M$\textit{:}$\,M\geq 2$, in the further text of the paper, is functional related to the fundamental parameters of internal geometry of a space, more exactly, to components of \textit{% Riemann-Christoffel's} curvature tensor. At the end, analogously to a notion of lines of a curvature in the differential geometry, the notion of sub-spaces of curvature of \textit{Riemannian} hyper-dimensional vector spaces is also generally defined.