Computational methods for calulating the flow in turbomachines, which have been developed over the last fifteen years, are reviewed and the development of a new method is described in this thesis. The new method enables calculations of transonic flows on the throughflow plane to be made for the first time and is based on the opposed difference form of the time marching numerical technique. The throughflow method has also been combined with a quasi-three dimensional blade-to-blade program and interactions between the two surfaces have been performed in an attempt to calculate the fully three dimensional transonic flow in turbomachinery. Comparisons with a full three dimensional time marching method for the transonic flow through a low hub:tip radius ratio steam turbine nozzle have shown that the iterative method can calculate three dimensional transonic flows accurately. From the results of these calculations it is shown that design methods should include an accurate estimate of the streamtube thickness distribution through blade rows. It was found that the amount of previous experimental work on three dimensional transonic flows in steam turbines was very sparse. An annular cascade was designed and built both as a check on theoretical predictions and also to provide a facility for the fundamental study of this type of flow. Comparisons between theory and experiment indicate that, for this particular case, the inviscid calculation methods predict the flow accurately and that significant viscous effects are confined to a small area near the blade tip. A semi-implicit numerical algorithm has also been developed in an attempt to speed up the time marching method and modest improvements have been obtained. Finally, conclusions are drawn from the work and suggestions made for possible future experimental and theoretical developments.