We analyze a multidimensional screening model in which a principal offers a menu of quality-price pairs to a consumer with multiple dimensions of private information and a quasilinear utility function. We derive necessary conditions for optimality, and use them to provide insight into optimal exclusion, positive trade, and screening. We then recast the problem in terms of incremental quality levels and prices, the so-called demand-profile approach (DPA). Under DPA, the problem decouples across increments and can be solved one at a time. We provide novel conditions under which DPA recovers the solution to the full problem exactly or approximately, and which make the necessary conditions sufficient for optimality: essentially, valuations must be sufficiently correlated across quality increments. Applied to empirical estimates of demand for health insurance, we show that DPA is approximately valid, and we apply it to understand equilibrium outcomes in a monopoly insurance market.