The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the two-locus case. Epistasis has been linked to biological important properties such as the advantage of sex. Gene interactions among multiple loci are usually reduced to two-way interactions. Here, we present a geometric theory of shapes of fitness landscapes for multiple loci. We investigate the dynamics of evolving populations on fitness landscapes and the predictive power of the geometric shape for the speed of adaptation. Finally, we discuss applications to fitness data from viruses and bacteria.