The authors investigate the \textsl{Yamabe flow } on a compact Riemannian manifold \((M^n,g_0)\): \[ \partial_tu=u^{4/(n-2)}\left( 4\frac{n-1}{n-2} \Delta_0u-R_0u\right), \] here \(R_0\) and \(\Delta_0\) are the scalar curvature and Laplace-Beltrami operator of \(g_0\) respectively. The existence of this flow for all times \(t\) was established by \textit{R. Hamilton} [Lectures on geometric flows. (1989), unpublished]; moreover, it was demonstrated that the flow converges exponentially fast to a Yamabe metric if \(R_0\) is negative [Hamilton (loc. cit.)], see also \textit{R. Ye} [J. Differ. Geom. 39, 35--50 (1994; Zbl 0846.53027), \textit{B. Chow} [Commun. Pure. Appl. Math. 45, 1003--1014 (1992; Zbl 0785.53027)]. Actually, the authors improve Hamilton's results. First, a Kazdan-Warner type condition which implies the convergence of the flow is found [cf. \textit{J. L. Kazdan} and \textit{F. W.Warner}, J. Differ. Geom. 10, 113--134 (1975; Zbl 0296.53037)]. Then it is demonstrated that an arbitrary compact manifold without boundary satisfies this quite natural condition if either \(3\leq n \leq 5\) or \((M^n,g_0)\) is locally conformally flat. Finally, the convergence of the Yamabe flow and the existence of a Yamabe metric is proved for \((M^n,g_0)\), \(3\leq n \leq 5\), with positive scalar curvature \(R_0\) and with positive Yamabe invariant \(Y_0\) [cf. R. Ye (loc. cit.)].