This is a sequel to Part I [J. Differ. Equations 120, No. 2, 319-410 (1995; Zbl 0827.53039)] which studies the prescribing scalar curvature problem on \(S^n\). First we present some existence and compactness results for \(n= 4\). The existence result extends those of \textit{A. Bahri} and \textit{J. M. Coron} [J. Funct. Anal. 95, No. 1, 106-172 (1991; Zbl 0722.53032)], \textit{M. Benayed}, \textit{Y. Chen}, \textit{H. Chtioui} and \textit{M. Hammami} [Duke Math. J. 84, No. 3, 633-677 (1996)] and \textit{D. Zhang} [New results on geometric variational problems, Thesis, Stanford Univ. (1990)]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions which, as a consequence, gives a complete description of when and where blow-ups occur. It follows from our results that solutions to the problem may have multiple blow-up points. This phenomenon is new and very different from the lower-dimensional cases \(n= 2,3\). Next we study the problem for \(n\geq 3\). Some existence and compactness results have been given in [the author, loc. cit.] when the order of flatness at critical points of the prescribed scalar curvature functions \(K(x)\) is \(\beta\in (n- 2, n)\). The key point there is that for the class of \(K\) mentioned above we have completed \(L^\infty\) a priori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of \(K(x)\) is \(\beta= n- 2\), the \(L^\infty\) estimates for solutions fail in general. In fact, two or more blow-up points occur. On the other hand, we provide some existence and compactness results when the order of flatness at critical points of \(K(x)\) is \(\beta\in [n- 2, n)\). With this result, we can easily deduce that \(C^\infty\) scalar curvature functions are dense in the \(C^{1, \alpha}\) \((0< \alpha< 1)\) norm among all positive functions. With respect to the \(C^2\) norm, such a density result is false in general. We also give a simpler proof to a Sobolev-Aubin type inequality established in [\textit{S.-Y. A. Chang} and \textit{P. C. Yang}, Duke Math. J. 64, No. 1, 27-69 (1991; Zbl 0739.53027)]. Some of the results in this paper as well as that of Part I have been announced in [C. R. Acad. Sci. Paris, Sér. I 317, No. 2, 159-164 (1993; Zbl 0787.53029)].