We propose a partial identification approach to estimate demand systems with market-level data when market shares are unobserved, but rankings of product sales are available. As is standard, we model consumer choice as arising from comparing product-specific utilities, which is the sum of systematic utilities (equal for all consumers) and idiosyncratic taste components. Market shares rankings restrict the feasible set of systematic utilities via demand inversion, which leads to moment inequalities that partially identify demand parameters. Our identified set is sharp under weak regularity, and the approach applies to a wide class of demand specifications, including random-coefficient models. The method accommodates endogenous product characteristics and can incorporate exact knowledge about some of the market shares for tighter identification. To compute confidence sets, we implement a computationally attractive procedure based on convex optimization for demand coefficients in the multinomial and nested logit models. In Monte Carlo simulations, our method systematically provides meaningful bounds with correct statistical coverage around the true demand parameters. In contrast, the leading approach in the literature of imputing market shares using power-law models before estimating demand results in substantial under-coverage of the true parameter.