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Multiphase and multidomain fluid flows associated with interfacial dynamics, steep jump in fluid properties, and moving boundaries between different phases and materials pose substantial computational challenges. In this paper, recent progresses made in numerical simulation using continuum and lattice–Boltzmann methods are highlighted, with special attention paid to issues related to time varying geometries, adaptive grid refinement, interfacial and geometric tracking, topological changes, and reduced numerical dispersion and dissipation. Illustrative physical applications including: (1) multiple rising bubbles interacting with carrier phase and with each other; (2) wave reflected on an inclined wall; and (3) the Rayleigh–Taylor instability problem, are presented to highlight the various aspects of the techniques developed. [ABSTRACT FROM AUTHOR]