Maximum Entropy data processing techniques have been widely available for use by NMR spectroscopisis and mass spectrometrisls since they were first reported as a tool for enhancing damaged images. However, the techniques have been met with a certain amount of scepticism amongst the spectroscopic community; not least their apparent ability to get something for nothing. The aim of the work presented in this thesis is to demonstrate that if these techniques are used carefully and in appropriate situations a great deal of information can be extracted from both NMR and mass spectra. This has been achieved by using the Memsys5 and Massive Inference algorithms to process a range of NMR and mass spectra which suffer from some of the problems which are commonly encountered in spectroscopy, i.e. poor resolution, poor sensitivity, how to process spectra with a wide range of peak widths. The theory underlying the two algorithms is described simply and the techniques for selecting appropriate point spread functions are outlined. Experimental rather than simulated spectra are processed throughout. Throughout this work the Maximum Entropy results are freated with scepticism. A pragmatic approach is employed to demonstrate that the results are valid. It is concluded that the Maximum Entropy methods do have their place amongst the many other data processing strategies used by spectroscopists. If used correctly and in appropriate situations the results can be worth the investment in time needed to obtain a satisfactory result.