In this chapter, a discussion is given of the many ways in which data taken from rotating machines can be analyzed. The problem is that there are often a number of interrelated parameters and the task of the analyst (engineer) is to clarify the key relationships. Most, if not all, vibrational behavior of machines is related to shaft rotation speed, and the chapter begins with a discussion of the Fourier transform and then moves on to a study of various methods of presenting data where other parameters are involved. Under quasi-static conditions, it is shown how displacement data can be effectively analyzed to produce effective threshold signals for a monitoring system. Attention is then turned to transient data as this is rich in information at the cost of some mathematical complexity. The two difficulties are discussed, firstly that of resolving the vibration into shaft orders and secondly, relating this to quasi-static studies. The principal features arising in shaft orbit are then discussed together with alternative methods of data presentation. The chapter ends with an evaluation of the diagnostic process as applied to machines.