Multifractal analysis, that quantifies the fluctuations of regularities in time series or textures, has become a standard signal/image processing tool. It has been successfully used in a large variety of applicative contexts. Yet, successes are confined to the analysis of one signal or image at a time (univariate analysis). This is because multivariate (or joint) multifractal analysis remains so far rarely used in practice and has barely been studied theoretically. In view of the myriad of modern real-world applications that rely on the joint (multivariate) analysis of collections of signals or images, univariate analysis constitutes a major limitation. The goal of the present work is to theoretically ground multivariate multifractal analysis by studying the properties and limitations of the most natural extension of the univariate formalism to a multivariate formulation. It is notably shown that while performing well for a class of model processes, this natural extension is not valid in general. Based on the theoretical study of the mechanisms leading to failure, we propose alternative formulations and examine their mathematical properties.