This thesis examines the effects of resource constraints on amplify-and-forward (AF) networks. Chapters 3 and 4 are the first research chapters. Chapter 3 studies the outage probability performance of a two-hop two-way AF peak power constrained orthogonal frequency division multiplexing (OFDM) network. Its performance is then optimized. Chapter 4 focuses on the one-way special case of the system studied in Chapter 3. It begins with an analysis of the network when nonlinear distortion produced by signal clipping dominates the additive noise in the system. To conclude Chapter 4, the theoretical study performed throughout Chapters 3 and 4 is used to optimize the performance of a one-way real world test bed. Chapter 5 studies the n-hop multiple-input multiple-output (MIMO) AF relay network. Novel techniques are developed using random dynamical system (RDS) theory and Lyapunov exponents to establish capacity and power scaling laws for the network as n grows large. One of the main conclusions is that the average transmit power must grow at an exponential rate if capacity decay across the network is to be avoided. Chapter 6 constitutes the final research chapter. In it, the techniques used to study peak-power constrained OFDM-based networks are combined with those developed in Chapter 5, which were used to study capacity and power scaling for multihop AF networks. The conclusion of this is that incorporating OFDM into peak-power constrained multihop AF relay networks will cause the capacity along each of the network's eigenchannels to decay exponentially. Finally, we show that the effects of distortion can be circumvented by ensuring the number of antennas at each node scales at a super-linear rate with the number of hops within the network.