We propose a general Bayesian approach to modelling epidemics such as COVID-19. The approach grew out of specific analyses conducted during the pandemic, in particular, an analysis concerning the effects of non-pharmaceutical interventions (NPIs) in reducing COVID-19 transmission in 11 European countries. The model parameterises the time-varying reproduction number R_t through a multilevel regression framework in which covariates can be governmental interventions, changes in mobility patterns, or other behavioural measures. Bayesian multilevel modelling allows a joint fit across regions, with partial pooling to share strength. This innovation was critical to our timely estimates of the impact of lockdown and other NPIs in the European epidemics: estimates from countries at later stages in their epidemics informed those of countries at earlier stages. Originally released as Imperial College Reports, the validity of this approach was borne out by the subsequent course of the epidemic. Our framework provides a fully generative model for latent infections and derived observations, including deaths, cases, hospitalizations, ICU admissions, and seroprevalence surveys. In this article, we additionally explore the confounded nature of NPIs and mobility. Versions of our model were used by New York State, Tennessee, and Scotland to estimate the current epidemic situation and make policy decisions.