The interpretation of experiments is complicated when the outcome of an experimental unit depends not only on its assigned treatment but also on interference from other units. Here, we extend the potential outcomes framework of causal inference without such interference between units (Rubin, 1974) in order to define and assess causal effects. When two units cannot interfere with each other, then one unit's treatment assignment only affects that unit's outcome. However, when two units can interfere with each other, then one unit's treatment assignment generally affects both of their outcomes. Furthermore, the interference can depend on units' characteristics and the treatment assignment itself, and is often only partially revealed. Our analysis of data generated by such situations uses both Bayesian and frequentist ideas to test sharp null hypotheses about causal effects. In particular, to assess causal effects we model and estimate the interference between units as a network, and develop novel testing procedures that involve repeated sampling of the treatment assignment under constraints from the network topology and the tested hypothesis. We illustrate our causal framework in applications where such forms of interference are ubiquitous but currently not adequately addressed.