A survey is made of progress in the development and application of multidimensional methods for predicting the in-cylinder flows of reciprocating engines: also discussed is the relevance to combustion prediction. It is concluded that there have been significant advances in the numerical methodology, such that it is now possible to contemplate calculation of the complex three-dimensional flows found in production engine combustion chambers. The cost and effort involved is, however, likely to be high for adequate computational grid densities, and there remain some unresolved problems in the representation of inlet and exhaust flows. A second important conclusion is that as the accuracy of the numerical methods has improved, so has the agreement between predictions and measurements of the ensemble-average flow based on conventional turbulence models. Thirdly, the accurate prediction of the flow behavior during individual cycles, which is of interest in connection with cycle-to-cycle variations in combustion performance, is believed still to be outside the capabilities of current methodology. It is argued, however, that advances in this direction are likely to be accelerated if it is recognized that similar phenomena occur in simple stationary flows, such as are found in stirred combustion bombs.