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The complex nature of wireless ad hoc networks requires fundamental research to address issues related to their modeling, performance analysis, design, and implementation. In the first part of this dissertation, we study the topological properties of wireless networks, abstracting them as random geometric graphs. We study how many neighbors each individual node needs to maintain in order to have certain desirable global properties. In order to guarantee being connected with probability going to one as the network size goes to infinity, we show that each node needs to connect to its Θ (log n) nearest neighbors. Specifically, 1.44 logn neighbors can guarantee asymptotic connectivity, while 0.074 logn leads to disconnectivity. We next study the θ-coverage of wireless networks of n nodes, with each node connecting to its ϕ n nearest neighbors. The network is called θ-covered if every node, except those along the …