In this thesis, the problem of estimating trust in distributed systems is considered. Distributed systems can be virtual or real-world systems in which multiple agents interact. One of the biggest problems in distributed systems is that they only function well if everyone contributes some resources. If agents do not participate, or try to pretend they participate, but in fact are slacking off, then the system might not achieve its desired purpose. This work presents two methods to estimate how well agents contribute to the network, while preventing cheating. Firstly, the NetFlow algorithm, which uses max-flow computations to bound the profit of cheating by Sybil attacks. Secondly, the Temporal PageRank algorithm, which uses information about the ordering of the interactions to provide a robust mechanism to determine reputation. Theoretical guarantees about several aspects of these algorithms are provided. Furthermore, these mechanisms are tested on data from a real-world distributed system: Tribler, an anonymous peer-to-peer downloading system. Finally, taking both the theoretical and practical results, the broader implications of these mechanisms and future possibilities are explored.