Summary: The concept of a Riemann-Cartan manifold was introduced by E. Cartan. A Riemann-Cartan manifold is a triple \((M,g,\bar\nabla)\), where \((M,g)\) is a Riemann \(n\)-dimensional \((n\geq2)\) manifold with linear connection \(\bar\nabla\) having nonzero torsion \(\bar S\) such that \(\bar\nabla g=0\). In our paper, we consider the scalar and total scalar curvatures of the Riemann-Cartan manifold \((M,g,\bar\nabla)\) and proved some formulas connecting these curvatures with the scalar and total scalar curvatures of the Riemannian \((M,g)\). In particular we analyze these formulas for the case of Weitzenbök manifolds. We also prove some vanishing theorems.