Medium Access Control (MAC) protocols deal with the problem of effectively utilizing wireless communication channels. Therefore, efficient MAC schemes are of great importance to achieving good performance of wireless networks. In this thesis, two types of techniques that can be utilized to improve the performance of the MAC schemes and potentially significantly improve the throughput of the wireless network are studied: power control and full-duplex communication.For any wireless communication system, it is critical to have an energy efficient MAC scheme given the limited power resource. This raises the question of how power control algorithm can help with the design of the MAC scheme. To answer this question, the classic Foschini-Miljanic power control algorithm is analyzed from a geometric perspective. A novel tool, the Meobius transform, is utilized for the first time to study the convergence condition of the power control algorithm. Based on the analysis, we propose an energy efficient MAC scheme that greatly improves the spatial reuse of the wireless networks. Moreover, the effects of the power constraints and fading on the convergence of the power control algorithms are also considered in the context of random networks.While focusing on the quest for a good MAC scheme that works for current wireless systems, an alternative way is to think outside the box and design a wireless system that is fundamentally different from the current communication system model. Half duplex operation, which means that a node cannot transmit and receive simultaneously over the same frequency band due to overwhelming self-interference, makes the MAC design in wireless medium more challenging than in wireline networks. This thesis challenges the paradigm by proposing a comprehensive investigation of the physical and MAC layers of virtual full-duplex networks based on rapid on-off-division duplex (RODD). The key to RODD is to allow all nodes to transmit and receive simultaneously by letting each node transmit through the on-slots of its on-off mask and receive over the off-slots. In case of a point-to-point connection of two nodes, both nodes can simultaneously transmit a message and receive a message, achieving full duplex at the frame level. Part of this thesis is on the design and prototype implementation of such virtual full-duplex wireless networks using software-defined radios and evaluating its performance. Novel channel coding, synchronization and estimation algorithms are designed for the RODD scheme to correctly recover the received on-off signal.Another type of full-duplex is achieved by canceling the self-interference caused by one's own transmitter. This thesis also investigates the throughput for wireless networks with full-duplex radios using stochastic geometry. Full-duplex (FD) radios can exchange data simultaneously with each other. On the other hand, the downside of FD transmission is that it inevitably causes extra interference to the network compared to half-duplex (HD) transmission. Moreover, the residual self-interference has negative effects on the network throughput. In this thesis, we focus on a wireless network of nodes with both HD and FD capabilities and derive and optimize the throughput in such a network. Our analytical result shows that if an ALOHA protocol is used, the maximal throughput is achieved by scheduling all concurrently transmitting nodes to work in either FD mode or HD mode depending on one simple condition. Moreover, the effects of imperfect self-interference cancellation on the signal-to-interference ratio (SIR) and throughput are also analyzed based on our mathematical model. We rigorously quantify the impact of imperfect self-interference cancellation on the throughput gain, transmission range, and other metrics, and we establish the minimum amount of self-interference suppression needed for FD to be beneficial.