The development of the theory and application of Monte Carlo Markov Chain methods, vast improvements in computational capabilities and emerging software alternatives have made it possible for more frequent use of Bayesian methods in reliability applications. Bayesian methods, however, remain controversial in Reliability (and some other applications) because of the concern about where the needed prior distributions should come from. On the other hand, there are many applications where engineers have solid prior information on certain aspects of their reliability problems based on physics of failure or previous experience with the same failure mechanism. For example, engineers often have useful but imprecise knowledge about the effective activation energy in a temperature-accelerated life test or about the Weibull shape parameter in the analysis of fatigue failure data. In such applications, the use of Bayesian methods is compelling as it offers an appropriate compromise between assuming that such quantities are known and assuming that nothing is known. In this paper we compare the use of Bayesian methods with the traditional maximum likelihood methods for a group of examples including the analysis of field data with multiple censoring, accelerated life test data, and accelerated degradation test data.