We establish a higher-dimensional version of multifractal analysis for hyperbolic flows. This means that we consider simultaneously the level sets of several Birkhoff averages. Examples are the Lyapunov exponents as well as the pointwise dimension and the local entropy of a given measure. More precisely, we consider multifractal spectra associated to multi-dimensional parameters, obtained by computing the entropy of the level sets associated to the Birkhoff averages. We also consider the more general class of flows with upper semicontinuous metric entropy. The multifractal analysis is obtained here from a variational principle for the topological entropy of the level sets, showing that their topological entropy can be arbitrarily approximated by the entropy of ergodic measures. This principle unifies many results. An analogous principle holds for the Hausdorff dimension. The applications include the study of the regularity of the spectra, the description of how these vary under small perturbations, and the detailed study of the finer structure. The higher-dimensional spectra also exhibit new nontrivial phenomena absent in the one-dimensional multifractal analysis.