The importance of time invariant, but unobserved characteristics of workers and their employers in the determination of wages is well known. This suggests that models of (un)employment featuring permanent worker and firm types are crucial for our understanding of labor markets. While such models are numerous, it was thought that even very simple ones were fundamentally not identifiable with wage data alone. Hence, the empirical content of these models were largely a mystery. With that, our understanding of permanent unobserved heterogeneity has been restricted to statistical models with limited economic interpretation. In the first chapter of my dissertation (joint with Marcus Hagedorn and Iourii Manovskii), I overcome this fundamental problem. I assess the empirical content of equilibrium models of labor market sorting based on unobserved (to economists) characteristics. Specifically, I develop quantitative tools to identify and estimate a wide class of models of labor market sorting. I not only find that many models are likely completely identifiable but that reliable estimates of key model primitives can be obtained using routinely available matched employer-employee datasets. In the second chapter of my dissertation (with Kory Kantenga), I apply the framework developed in the first chapter to study the role of worker and employer heterogeneity in driving German wage inequality between 1993 and 2007. The model I earlier developed fits overall wage variance, the wage function, search frictions, unemployment levels and the degree of sorting between workers and firms. The fit of the model is comparable to non-structural approaches which utilize many more degrees of freedom. In decomposing the rise in German wage inequality, I find that changes in the non-parametrically estimated production function and the sorting between worker and firm types that it induces accounts for most of the increase in German wage inequality. Finally, by testing whether log wage differences between employees who are coworkers at two distinct firms are constant, I show that the commonly assumed log additive separability approximation of log wages is rejected. Finally, the estimated model is capable of reproducing the degree of non additive separability in the data.