Orthogonal Frequency Division Multiplexing (OFDM) is being considered for high speed communication over wireless (acoustic) underwater channels which are characterized by frequency selectivity and strong motion-induced Doppler distortion. Receiver algorithms and adaptive power control are the focus of this thesis. Two types of receiver algorithms are considered: those based on coherent detection and those based on differentially coherent detection. While coherent detection theoretically offers better performance, it relies on accurate channel estimation and phase tracking, which are challenging in conditions of fast time variations. Coherent detection then suffers, making differentially coherent detection a preferred choice in many situations. To partially offset for the channel variation, adaptive power control based on receiver feedback is considered. Proof of concept of various communication techniques is usually left to experimental testing because of the lack of widely accepted statistical models for underwater acoustic channels. Real data recordings from a recent Mobile Acoustic Communication Experiment (MACE-2010) are used to compare the performance of adaptive receiver algorithms. In addition, an in-air acoustic test-bed has been developed as a part of this thesis. Such an approach is deemed to provide a more realistic vision of the channel conditions than a simplified computer simulation, while overcoming the cost and difficulty of at-sea deployments. Processing of the in-air acoustic data confirms the similarity with underwater conditions. Moreover, in-air tests enable on-line demonstration of adaptive power control.