Abstract Recent advances in Global Positioning System (GPS), tilt and Interferometric Synthetic Aperture Radar (InSAR) have greatly increased the availability of volcano deformation data. These measurements, combined with appropriate source models, can be used to estimate magma chamber depth, and to provide information on chamber shape and volume change. However, kinematic models cannot constrain magma chamber volume, and provide no predictive capability. Volcanic eruptions are commonly preceded by periods of inflation. Under appropriate conditions, eruptions are ‘inflation predictable’; that is, subsequent eruptions occur when inflation recovers the deflation during the preceding event. Notable successes in forecasting eruptions have come largely through the ability to discern repeatable patterns in seismic activity, ground deformation and gas emission, combined with historical and geological evidence of past eruptive behaviour. To move beyond empirical pattern recognition to forecasting based on deterministic physical–chemical models of the underlying dynamics, will require integration of different data types and models. I suggest two areas poised for progress: quantitative integration of deformation and seismicity; and model-based forecasts conditioned on estimates of material parameters and initial conditions from inversion of available datasets. Deformation and seismicity are the principal geophysical methods for volcano monitoring, and in some cases have signalled dyke propagation minutes to hours prior to eruptions. Quantitative models relating these processes, however, have been lacking. Modern theories of seismicity rate variations under changing stress conditions can be used to integrate deformation and (volcano–tectonic) seismicity into self-consistent inversions for the spatio-temporal evolution of dyke geometry and excess magma pressure. This approach should lead to improved resolution over existing methods and, perhaps, to improved real-time forecasts. The past few decades have also witnessed a marked increase in the sophistication of physical–chemical models of volcanic eruptions. I review conduit models that can be combined with GPS and extrusion rate data through Markov Chain Monte Carlo (MCMC) inversion to estimate the absolute volume of the crustal magma chamber, initial chamber overpressure, initial volatile concentrations and other parameters of interest. The MCMC estimation procedure can be extended to deterministic forecasting by using the distribution of initial conditions and material parameters consistent with available data to initiate predictive forward models. Such physics-based MCMC forecasts would be based on all knowledge of the system, including data up to the current date. The underlying model is completely deterministic; however, because the method samples initial conditions and physical parameters consistent with the given data, it yields probabilistic forecasts including uncertainties in the underlying parameters. Because there are almost certain to be effects not factored into the forward models, there is likely to be a substantial learning curve as models evolve to become more realistic.