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Models of physical phenomena such as crystal growth or blood flow generally involve complex moving interfaces, with velocities determined by interfacial geometry and material physics. Numerical methods for such models tend to be customized. As a consequence, they must be redesigned whenever the model changes. We present a general computational algorithm for evolving complex interfaces which treats the velocity as a black box, thus avoiding model-dependent issues. The interface is implicitly updated via an explicit second-order semi-Lagrangian advection formula which converts moving interfaces to a contouring problem. Spatial and temporal resolutions are decoupled, permitting grid-free adaptive refinement of the interface geometry. A modular implementation computes highly accurate solutions to geometric moving interface problems involving merging, anisotropy, faceting, curvature, dynamic topology and nonlocal interactions. Coupled with fast new boundary integral schemes for elliptic partial differential systems, the implementation provides fast accurate solutions of viscoelastic and creeping flows with complex interfaces.