Progress is usually a sequence of events where advances in one field open up new opportunities in another, which in turn makes it possible to push yet another field forward, and so on. Thus, the development of fast and powerful computers has led to the development of new numerical methods for direct numerical simulations (DNS) of multiphase flows that have produced detailed studies and improved knowledge of multiphase flows. While the origin of DNS of multiphase flows goes back to the beginning of computational fluid dynamics in the early sixties, it is only in the last decade and a half that the field has taken off. We, the authors of this book, have had the privilege of being among the pioneers in the development of these methods and among the first researchers to apply DNS to study relatively complex multiphase flows. We have also had the opportunity to follow the progress of others closely, as participants in numerous meetings, as visitors to many laboratories, and as editors of scientific journals such as the Journal of Computational Physics and the International Journal of Multiphase Flows. To us, the state of the art can be summarized by two observations:

• Even though there are superficial differences between the various approaches being pursued for DNS of multiphase flows, the similarities and commonalities of the approaches are considerably greater than the differences. […]