Summary: Let \((S^ n,g_ 0)\) be the standard \(n\)-sphere. The following question was raised by L. Nirenberg. Which function \(K(x)\) on \(S^ 2\) is the Gauss curvature of a metric \(g\) on \(S^ 2\) conformally equivalent to \(g_ 0\)? Naturally one may ask a similar question in higher dimensional case, namely which function \(K(x)\) on \(S^ n\) is the scalar curvature of a metric \(g\) on \(S^ n\) conformally equivalent to \(g_ 0\)? We give some a priori estimates for solutions of the prescribing scalar curvature equations for \(n \geq 3\) and some existence results which are quite natural extensions of previous results of \textit{A. Chang} and \textit{P. Yang} [Acta Math. 159, 215-259 (1987; Zbl 0636.53053)] and \textit{A. Bahri} and \textit{J. M. Coron} [J. Funct. Anal. 95, 106-172 (1991; Zbl 0722.53032)] for \(n = 2,3\). Details and more general results will be given in [the author, Prescribing scalar curvature on \(S^ n\) and related problems, parts I and II (in preparation)]. For \(n = 3\) such estimates have been obtained by R. Schoen and D. Zhang [see \textit{R. Schoen}, Private notes of special topics in geometry courses in Stanford University and New York University 1988 and 1989; \textit{D. Zhang}, New results on geometric variational problems, Thesis, Stanford University (1990)]. As a byproduct of the blow up analysis completed here for \(n \geq 3\) we have essentially extended all the results in [the author, Commun. Pure Appl. Math. 46, No. 3, 303-340 (1993), C. R. Acad. Sci., Paris, Sér. I 314, No. 1, 55-59 (1992; Zbl 0744.53023) and J. Funct. Anal. (to appear)] for \(n = 3,4\) to higher dimensional cases. More precisely, we show that for the prescribing scalar curvature problem on \(S^ n\), we can perturb (in an explicit way) any given positive continuous function in any neighborhood of any given point on \(S^ n\) such that for the perturbed function there exist many solutions. The related critical exponent equations \(-\Delta u = K(x)u^{(n+2)/(n-2)}\) in \(\mathbb{R}^ n\) with \(K(x)\) being periodic in one of the variables are also studied and infinitely many positive solutions (modulo translations by its periods) are obtained under some additional mild hypotheses on \(K(x)\).