In this work, we study a model of the interaction of n species (n >2) which involves cross-diffusion. The model generalizes the model introduced in 1979 by Shigesada, Kawasaki and Teramoto for two species (SKT). All species are assumed to exhibit a functional response of the same form similar to SKT model. By constructing a Lyapunov functional of the system subject to some conditions on the cross-diffusion matrix and the diffusion vector, we established the global stability of the constant equilibrium. A sufficient condition is also derived for the coexistence of a large number of interacting species. We also established the uniform W₂¹  estimate of the solution for the the generalised SKT model in the process of which the existence of global solution is proved. Particular cases of the model for two species are considered extensively in the literature. Most of these results are shown to follow as consequences of the general theory developed here.