A system is fault-tolerant if it maintains some form of correctness in the presence of faults. Fault-tolerant systems are necessary in many application areas of computing, especially where the failure of a computer system may lead to considerable damage. However, the development of fault-tolerant systems is a difficult task and requires rigorous and scientifically sound engineering methodologies. Amoung other topics, these methodologies must cover (1) system and fault modelling, (2) design patterns for fault-tolerance mechanisms and (3) basic building blocks for fault-tolerant algorithms. This thesis contributes to research in all of these three areas. Questions of system and fault modelling are treated in chapter 2 in which the four most prominent system models for fault-tolerant distributed systems are presented and compared: the model of partial synchrony by Dwork, Lynch and Stockmeyer, the failure detector model of Chandra and Toueg, the timed asynchronous system model of Cristian and Fetzer, and the quasi-synchronous model by Almeida, Verissimo and Casimiro. Chapter 3 focusses on the underlying principles of fault-tolerant systems. Starting point is the fault-tolerance theory of Arora and Kulkarni. In this theory, a fault-tolerant program is regarded as the composition of a fault-intolerant program and fault-tolerance components. We analyse the assumptions of the theory and extend it by defining formal notions of redundancy. Two forms of redundancy are identified (redundancy in space and redundancy in time) and we study which forms of redundancy are necessary to maintain which properties in the presence of faults. In chapter 4 we develop algorithmical building blocks for observation in faulty environments, a central problem in fault-tolerant computing which has not enjoyed a lot of research attention yet. Observation here means detecting whether or not a boolean predicate on global states holds during the execution of a distributed system. In the asynchronous system model with crash failures, we introduce two new observation modalities called `negotiably' and `discernibly' (which roughly correspond to the well-known modalities `possibly' and `definitely' by Cooper, Marzullo and Neiger) and present detection algorithms for them under increasingly weak fault assumptions.