Wireless communications systems are used today in a variety of milieux, with a recurring theme: users and applications regularly require higher throughput. Multiple antennas enable higher throughput and/or more robust performance than single-antenna communications, with no increase in power or frequency bandwidth. Systems are required which achieve the full potential of this "space-time" communication channel under the significant challenges of time-varying fading, multiple users, and the choice of appropriate coding schemes. This dissertation is focused on solutions to these problems. For the single-user case, there are many well-known coding techniques available; in the first part of this dissertation, the performance of two of these methods are analyzed. Trained and differential modulation are simple coding techniques for single-user time-varying channels. The performance of these coding methods is characterized for a channel having a constant specular component plus a time-varying diffuse component. A first- order auto-regressive model is used to characterize diffuse channel coefficients that vary from symbol to symbol, and is shown to lead to an effective SNR that decreases with time. A lower bound on the capacity of trained modulation is found for the specular/diffuse channel. This bound is maximized over the training length, training frequency, training signal, and training power. Trained modulation is shown to have higher capacity than differential coding, despite the effective SNR penalty of trained modulation versus differential methods. The second part of the dissertation considers the multi-user, multi-antenna channel, for which capacity-approaching codes were previously unavailable. Precoding with the channel inverse is shown to provide capacity that approaches a constant as the number of users and antennas simultaneously increase. To overcome this limitation, a simple encoding algorithm is introduced that operates close to capacity at sum-rates of tens of bits/channel-use. The algorithm is a variation on channel inversion that regularizes the inverse and uses a "sphere encoder" to perturb the data to reduce the energy of the transmitted signal. Simulation results are presented which support our analysis and algorithm development.