Complex networks have been receiving increasing attention by the scientific community, thanks also to the increasing availability of real-world network data. So far, network analysis has focused on the characterization and measurement of local and global properties of graphs, such as diameter, degree distribution, centrality, and so on. In the last years, the multidimensional nature of many real world networks has been pointed out, i.e. many networks containing multiple connections between any pair of nodes have been analyzed. Despite the importance of analyzing this kind of networks was recognized by previous works, a complete framework for multidimensional network analysis is still missing. Such a framework would enable the analysts to study different phenomena, that can be either the generalization to the multidimensional setting of what happens in monodimensional networks, or a new class of phenomena induced by the additional degree of complexity that multidimensionality provides in real networks. The aim of this paper is then to give the basis for multidimensional network analysis: we present a solid repertoire of basic concepts and analytical measures, which take into account the general structure of multidimensional networks. We tested our framework on different real world multidimensional networks, showing the validity and the meaningfulness of the measures introduced, that are able to extract important and non-random information about complex phenomena in such networks. © 2012 Springer Science+Business Media, LLC.