Analytical modeling of turbomachinery components and systems has been used for more than a century to develop new machines and understand internal flow states. Flow modeling basics are reviewed in this survey including a summary of the flow processes observed in nature. The development of a variety of different loss, blockage, and deviation models is reviewed, and the complexity of mathematical data processing and model development is presented. Examination of different modeling philosophies is given with critique of the consequences. Examples of data matching, modeling for design work, and modeling uncertainty are given. Suggestions for future improvements are offered.