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In this thesis, we study wireless communications and cognitive radio transmissions under quality of service (QoS) constraints and channel uncertainty. Initially, we focus on a time-varying Rayleigh fading channel and assume that no prior channel knowledge is available at the transmitter and the receiver.  We investigate the performance of pilot-assisted wireless transmission strategies. In particular, we analyze different channel estimation techniques, including single-pilot minimum mean-square-error (MMSE) estimation, and causal and noncausal Wiener filters, and analyze efficient resource allocation strategies. Subsequently, we study the training-based transmission and reception schemes over a priori unknown, Rayleigh fading relay channels in which the fading is modeled as a random process with memory. In the second part of the thesis, we study the effective capacity of cognitive radio channels in order to identify the performance in the presence of statistical quality of service (QoS) constraints. The cognitive radio users are assumed to initially perform channel sensing to detect the activity of primary users and then transmit the data at two different average power levels depending on the presence or absence of active primary users. We conduct the performance analysis in both single-band and multi-band environments in the presence of interference constraints. Later, we consider a cognitive radio system in which the cognitive secondary users operate under channel uncertainty and QoS constraints, and perform both channel estimation and sensing. In this setting, we analyze the effective capacity and determine efficient power and rate allocation policies. Finally, we study cognitive multiple-input multiple-output (MIMO) channels in the low-power regime, and investigate the energy efficiency.