Reliability analysis is an important application area of statistics and probability theory in engineering, with several specific features which often complicate application of standard methods. Such features include data censoring, for example due to maintenance activities, lack of knowledge and information on dependence between random quantities, for example if failures occur due to competing risks, and required use of expert judgements, for example when new or upgraded versions of units are used. While mathematical approaches for dealing with such issues have been presented within the framework of statistics using precise probabilities, the use of imprecise probability provides exciting new ways for dealing with such challenges in reliability. One of the first approaches that generalized probability in reliability was fuzzy reliability theory, but this suffers from vagueness about axioms and rules for combination of information, and lack of clear interpretation of the results. We restrict attention to generalized uncertainty quantification in reliability via lower and upper probabilities, also known as ‘imprecise probability’ or ‘interval probability’. During the last two decades, imprecise probability has received increasing attention, and interesting applications have been reported. It is widely accepted that, by generalizing precise probability theory in a mathematically sound manner, with clear axioms and interpretations, this theory provides a better approach to generalized uncertainty quantification then its current alternatives.