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A common feature of many free surface flows—drop/bubble breakup or coalescence and film/sheet rupture—is the occurrence of hydrodynamic singularities. Accurately computing such flows with continuum mechanical, multidimensional free surface flow algorithms is a challenging task given these problems’ multiscale nature, which necessitates capturing dynamics occurring over disparate length scales across 5–6 orders of magnitude. In drop breakup, the thinning of fluid threads that form and eventually pinch-off must be simulated until the thread's radius is about 10 nm. When two drops approach one another, the thickness of the fluid film separating them must fall below 10 nm before coalescence is said to have occurred. If the initial drop radii are 1 mm, simulations must remain faithful to the physics as thread radius or film thickness falls from 10−3 m to below 10−8 m. Here we review significant findings in interfacial flows with hydrodynamic singularities spearheaded by sharp interface algorithms. These multidimensional algorithms can achieve resolution that to date has only been possible with the use of simple 1D evolution equations. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 55 is January 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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