Explosive volcanic eruptions produce fragmented volcanic rocks, known as volcanic ash, which constitutes one of the most widespread volcanic hazards, and can be transported and deposited on international scales. Volcanic ash has impacts on infrastructure, agriculture, transport and human health. In this talk, I will describe the fluid dynamics of ash transport from source to deposit, and introduce some key features of volcanic processes and ash deposits. I will focus particularly on recent advances in modelling the transport of volcanic ash in the atmosphere, including fundamental insights into ash cloud dispersion, and operational approaches for assessing ash hazard to aviation.