Interference and collisions greatly limit the throughput of mesh networks that used contention-based MAC protocols such as 802.11. It is widely believed that significantly higher throughput is achievable if transmissions are scheduled. However, since the typical approach to this throughput optimization requires optimizing over a space that is exponential in the number of links, the optimal throughput has appeared to be computationally intractable for all but small networks. This research presents techniques that are typically able to efficiently compute optimal schedules as well as optimal routing. The technique consists of three layers of optimization. The inner-most optimization computes an estimate of the throughput. This optimization is a standard linear or nonlinear constrained optimization, depending on the objective function. The middle iteration uses the Lagrange multipliers from the inner-most optimization to modify the space over which inner-most optimization is performed. This is a graph theoretic optimization known as the maximum weighted independent set (MWIS) problem. The solvability of this problem is extensively studied, and the empirical evidence shows that usually MWIS arising from wireless mesh network can be solved quickly. The outer-most optimization uses the Lagrange multipliers from the inner-most optimization to find optimal routes. This optimization solves several least cost paths problems and several maximum weighted independent set problems. Together, these techniques allow optimal schedules to be computed for networks with hundreds and even a few thousand links, and allow optimal routes to be computed for networks with a few hundred links. Thus, the approach scales to the size of current and planned urban mesh networks. The techniques have been verified on networks generated with a realistic urban propagation simulator. In this setting, it is shown that optimal scheduling increases the throughput by a factor between three and ten over 802.11 CSMA/CA, depending on the density of the mesh routers and gateways. This thesis also studies the communication models. It is well known that the traditional protocol models have the drawback that they do not accurately model interference. Therefore, the actual throughput provided by these traditional protocol models is poor no matter how good the theoretical throughput offered. A general SINR protocol model is proposed to more accurately represent the interference. Furthermore, techniques were developed to overcome the interference from multiple sources. The ability to accommodate links that support multiple bit-rates and multiple transmit powers (e.g., 802.11) has been largely neglected by previous efforts on throughput maximization. Accommodating all possible bit-rates and transmission powers into the schemes developed in this thesis greatly increases the computational complexity. Thus, several heuristics are developed and examined.