Man has been fascinated by fish swimming for a long time, which has led to many scientific studies and debate. It was the pioneering paper by Gray (1936) some seventy years ago that led to the so-called Gray’s Paradox, which estimated the power required for a dolphin to swim at a constant speed to be several magnitudes larger than the power generated by its muscles, continues to be the subject of intense debate in the literature. The research during the past century has created a large body of knowledge on aquatic fish swimming, yet more important issues have to be addressed and are difficult to tackle with experiments alone. These include among others the three-dimensional structure of fish wakes, the effect of Reynolds number and/or approach flow conditions on swimming performance and energetics under controlled fish swimming and body motion parameters. It is an objective of this work to employ the computational tools we have developed to contribute toward addressing some of the aforementioned issues. Doing so is critical prerequisite not only from a pure biological standpoint but also in order to guide the design of biomimetically inspired vehicles and propulsive systems across a range of scales.