Physical phenomena in the Riemannian space of arbitrary dimensionality D= D' + 1 (D1 = integer) are analyzed. It is noted, that with the dimensionality of physical space of the observable Metagalaxy D = 3 + 1 very important physical features are connected, which distinguish that value of D from the others. However, this does not mean that the dimensionality of the physical space of the whole Universe must coincide with the value D = 3 + 1. This conclusion is confirmed by the geometric interpretation of supergravity and Kaluza-Klein theory. Different models of compactification and dimensional reduction of physical space and models of Metagalaxy formation in multidimensional spaces are reviewed.