Chikungunya is a viral disease transmitted to humans primarily via the bites of infected Aedes mosquitoes. The virus caused a major epidemic in the Indian Ocean in 2004, affecting millions of inhabitants, while cases have also been observed in Europe since 2007. We developed a stochastic spatiotemporal model of Aedes albopictus-borne chikungunya transmission based on our recently developed environmentally-driven vector population dynamics model. We designed an integrated modelling framework incorporating large-scale gridded climate datasets to investigate disease outbreaks on Reunion Island and in Italy. We performed Bayesian parameter inference on the surveillance data, and investigated the validity and applicability of the underlying biological assumptions. The model successfully represents the outbreak and measures of containment in Italy, suggesting wider applicability in Europe. In its current configuration, the model implies two different viral strains, thus two different outbreaks, for the two-stage Reunion Island epidemic. Characterisation of the posterior distributions indicates a possible relationship between the second larger outbreak on Reunion Island and the Italian outbreak. The model suggests that vector control measures, with different modes of operation, are most effective when applied in combination: adult vector intervention has a high impact but is short-lived, larval intervention has a low impact but is long-lasting, and quarantining infected territories, if applied strictly, is effective in preventing large epidemics. We present a novel approach in analysing chikungunya outbreaks globally using a single environmentally-driven mathematical model. Our study represents a significant step towards developing a globally applicable Ae. albopictus-borne chikungunya transmission model, and introduces a guideline for extending such models to other vector-borne diseases.