We present a randomization-based framework for estimating causal effects under interference between units. The framework integrates three components: (i) an experimental design that defines the probability distribution of treatment assignments, (ii) a mapping that relates experimental treatment assignments to exposures received by units in the experiment, and (iii) estimands that make use of the experiment to answer questions of substantive interest. Using this framework, we develop the case of estimating average unit-level causal effects from a randomized experiment with interference of arbitrary but known form. We establish conditions under which the resulting estimators, based on inverse probability weighting, are consistent and asymptotically normal. We provide randomization-based variance estimators that account for the complex clustering that can occur when interference is present. We illustrate and assess empirical performance with a naturalistic simulation based upon data from National Longitudinal Study of Adolescent Health. We provide an application to a randomized controlled trial on network diffusion of norms in 56 US schools.