This paper studies multi-dimensional matching between workers and jobs. Workers differ in manual and cognitive skills and sort into jobs that demand different combinations of these two skills. To study this multi-dimensional sorting problem, I develop a theoretical framework that generalizes the unidimensional notion of assortative matching and sufficient conditions on the technology under which sorting obtains. I derive the equilibrium in closed form and use this explicit solution to study biased technological change. The main finding is that an increase in worker-job complementarities in cognitive relative to manual inputs leads to more pronounced sorting and wage inequality across cognitive relative to manual skills. This can trigger wage polarization and boost aggregate wage inequality. I then estimate the model for the US and identify sizable technology shifts: during the last two decades, worker-job complementarities in cognitive inputs strongly increased whereas complementarities in manual inputs decreased. In addition to this bias in complementarities, there has been a cognitive skill-bias in production. Counterfactual exercises suggest that these technology shifts (as opposed to changes in skill supply and demand) can account for observed changes in worker-job sorting, wage polarization and a significant part of the increase in US wage dispersion.