Mosquito-borne diseases are spreading around the globe and frequently emerge in new regions. An increasing globalisation, urbanisation and climate change all contribute to this spread. And this spread is even worrying for the temperate UK: there is strong concern that vectors and diseases could arrive with a warming climate. The Asian tiger mosquito Aedes albopictus, able to transmit more than 20 different viruses to humans, has recently become established over large parts of Europe. It has already caused disease outbreaks in Italy, France, Croatia and Spain and its eggs have now been found in the UK too. In this work we present the development of several climate-driven models to analyse the spread and dynamics of this vector species. A range of regional and global climate data sets has been used to assess the mosquito's ability to become established, to spread and to transmit diseases in the UK and elsewhere. The models are validated against various mosquito recordings, mosquito abundance data and disease incidence data sets. We find that for now, only some densely populated and high importation risk areas in south-east England seem to be suitable for long-term Ae. albopictus survival. Future scenarios suggest that the mosquito could become established over larger parts of the UK though, spreading over Southern England within 30 years once introduced. We also use a laboratory study to analyse if European Ae. albopictus show any adaptation to colder climates that would allow the species to expand its range in Europe even further. Though we do find some differences in larval development in comparison to tropical strains, the results are not quite clear. Finally, we model arbovirus transmission by Ae. albopictus in countries where the species is endemic and, in a second step, apply findings to the UK. Results indicate that the UK is not yet warm enough to enable mosquito populations to build up to high numbers and effectively transmit arboviruses. Individual cases of disease transmission in summer months cannot be excluded though. General findings demonstrate the importance of using detailed vector and infection ecology when modelling mosquito-borne diseases, especially for temperate regions.