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We present a solution to the mesh tangling problem in surface tracking. Using an explicit triangle mesh to track the location of a surface as it moves in three dimensions has many potential advantages for accuracy and efficiency, compared to implicit capturing methods such as level sets. However, particularly when “mesh surgery” is required for topological changes, this approach is prone to tangling: The mesh may self-intersect or otherwise no longer represent a physical interface. Our new approach uses robust collision testing to determine when a mesh operation—such as motion, adaptive refinement, coarsening, or topological change—will lead to an invalid state; we then either roll back noncritical operations or apply robust collision response algorithms, minimally perturbing the mesh to guarantee validity. We present numerical examples demonstrating the robustness and accuracy of the method.