Abstract: Gromov proved the following ''Limit theorem'': Let g be a C^2 Riemannian metric on a smooth manifold M (without boundary). If a sequence of C^2 Riemannian metrics on M converges to g in the C^0 sense, and each scalar curvature is bounded from below by k. Then the scalar curvature of the limiting metric g is also bounded from below by k. In this talk, I'd like to talk about some total-scalar-curvature-version theorems of this limit theorem. I also consider the limit theorem for an weighted total scalar curvature and as a corollary, I give a definition of scalar curvature lower bound in a weak sense. To prove these, we use the Ricci flow. If I have time, I also would like to talk about limit theorems for the upper bound of the total scalar curvature. Compared to the above results, we use a different geometric flow: the Yamabe flow.