Flows with immiscible materials occur in many areas, including industrial and natural applications such as spray atomization, oil-water separation, ink-jet printing, cloud formation, and so on. When coupled with turbulence or fluid instabilities, the sharp interfaces between the immiscible materials undergo complex morphological changes, including interface break-up and reconnection, in addition to significant interface corrugation. How to accurately capture these phenomena is still an open question despite advances made with sharp interface methods such as volume of fluid, interface tracking or level sets methods. In general, these methods introduce specific interfacial properties, such as surface tension, into the governing equations in an ad-hoc manner. Furthermore, the discretized equations are regularized through the numerical algorithm, which further reduces accuracy and the methods generally struggle to maintain mass and momentum conservation. For example, in the volume-of-fluid method, the piecewise interface reconstruction has limited accuracy and the calculation of interface curvature from the discontinuous volume fraction, used to estimate surface tension, introduces significant error. Diffuse interface methods, such as Cahn-Hilliard-type approaches, remedy many of the issues of sharp interface methods, as the conservation laws are naturally satisfied and the interfacial properties arise from first principles, however important open questions remain related to the convergence to the sharp interface limit, numerical stability for high material property ratios, generalizations to, e.g., compressible multi-component flows, coupling with turbulence models, and the development of efficient adaptive methods to reduce computational cost while preserving accuracy. The minisymposium offers a venue to discuss the latest generalizations and applications of diffuse interface methods, as well as recent developments in robust numerical implementations.