The biological fitness of microbes is largely determined by the rate with which they replicate their biomass composition. Mathematical models that maximize this balanced growth rate while accounting for mass conservation, reaction kinetics, and limits on dry mass per volume are inevitably non-linear. Here, we develop a general theory for such models with full rank stoichiometric matrices, termed Growth Balance Analysis (GBA), which provides explicit expressions for protein concentrations, fluxes, and growth rates. These variables are functions of the concentrations of cellular components, for which we calculate marginal fitness costs and benefits that are related to metabolic control coefficients. At maximal growth rate, the net benefits of all concentrations are equal. For general models with rank-deficient stoichiometric matrices, we present the mathematical description of cellular growth states in terms of a decomposition into some limited number of growth modes, which are uniquely determined by the singular value decomposition (SVD) of the stoichiometric matrix.