The authors review and project their group's work in reaction engineering, electrokinetics, thin-film lubrication/wetting, biosensing, mass spectrometry, etc., that share one common mathematical underpinning: singularities. These can be geometric singularities of actual surfaces or objects, where focused electrical, acoustic, optical and shear-stress fields produce anomalous physical phenomena that have been explored mathematically with a spectral theory or exploited for specific applications. They are also singularities of mathematical manifolds, such as solution branches and Riemann manifolds, defined by abstract mathematical formulations, so that they can be used to design optical sensors, and understand nonlinear dynamical behavior that is relevant to system control and surface characterization. The common mathematical framework for these diverse topics underscores how mathematics can reveal, organize and inspire real and industrially relevant problems in chemical engineering. (c) 2013 American Institute of Chemical Engineers AIChE J, 59: 18301843, 2013