This report describes the Bayesian approach to probability theory with emphasis on the application to the evaluation of experimental data. A brief summary of Bayesian principles is given, with a discussion of concepts, terminology and pitfalls. The step from Bayesian principles to data processing involves major numerical efforts. We address the presently employed procedures of numerical integration, which are mainly based on the Monte Carlo method. The case studies include examples from electron spectroscopies, plasma physics, ion beam analysis and mass spectrometry. Bayesian solutions to the ubiquitous problem of spectrum restoration are presented and advantages and limitations are discussed. Parameter estimation within the Bayesian framework is shown to allow for the incorporation of expert knowledge which in turn allows the treatment of under-determined problems which are inaccessible by the traditional maximum likelihood method. A unique and extremely valuable feature of Bayesian theory is the model comparison option. Bayesian model comparison rests on Ockham's razor which limits the complexity of a model to the amount necessary to explain the data without fitting noise. Finally we deal with the treatment of inconsistent data. They arise frequently in experimental work either from incorrect estimation of the errors associated with a measurement or alternatively from distortions of the measurement signal by some unrecognized spurious source. Bayesian data analysis sometimes meets with spectacular success. However, the approach cannot do wonders, but it does result in optimal robust inferences on the basis of all available and explicitly declared information.