This paper studies the sorting of firms and directors appointed to their boards. I leverage a novel finite-mixture random-effects model to estimate the contribution of unobserved firm and director heterogeneity while being the first to explicitly allow for an interaction between the two to estimate the quality of the match between board members and firms. Results reveal that positive complementarities drive positive sorting. Using hand-collected data and textual analysis to build a large dataset on directors’ skills and qualifications, I find directors with specialized skill sets to be associated with higher complementarities while, consistent with the idea of knowledge hierarchies in the firm. On the contrary, CEOs or CFOs tend to be generalists relying on directors’ advice. Finally, I exploit unexpected deaths of board members to establish a positive causal effect of boards, where productivity is concentrated to a few highly complementary directors, on firm value and firm performance.