This paper derives conditions under which preferences and technology are nonparametrically identiﬁed in hedonic equilibrium models, where products are diﬀerentiated along more than one dimension and agents are characterized by several dimensions of unobserved heterogeneity. With products diﬀerentiated along a quality index and agents characterized by scalar unobserved heterogeneity, single crossing conditions on preferences and technology provide identifying restrictions. We develop similar shape restrictions in the multi-attribute case. These shape restrictions, which are based on optimal transport theory and generalized convexity, allow us to identify preferences for goods diﬀerentiated along multiple dimensions, from the observation of a single market. We thereby extend identiﬁcation results in Matzkin (2003) and Heckman, Matzkin, and Nesheim (2010) to accommodate multiple dimensions of unobserved heterogeneity. One of our results is a proof of absolute continuity of the distribution of endogenously traded qualities, which is of independent interest.