We study the identification and estimation of a multidimensional screening model, where a monopolist sells a multi-attribute product to consumers with private information about their multidimensional preferences. Under optimal screening, the seller designs product and payment rules that exclude "low-type" consumers, bunches the "medium types" at "medium-quality" products, and perfectly screens the "high types." Under the assumption that the cost function is quadratic and additively separable in products, we determine sufficient conditions to identify the joint distribution of preferences and the marginal costs from data on optimal individual choices and payments. Then, we propose estimators for these objects, establish their asymptotic properties, and assess their small-sample performance using Monte Carlo experiments.