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This book presents the state of the art of asymptotic and related mathematical methods, and how to apply them, as the means of choice, to representative building blocks of interfacial-flow phenomena. After an introduction which exemplifies the application of perturbation techniques in describing the well-known teapot effect, chapter 2 covers the status quo of the theory of inviscid sloshing and the associated modal analysis of free-surface waves; and chapter 3 envisages the intersection between dimensional analysis, scaling laws and the reduction of the governing partial differential equations to ordinary ones. The other chapters focus on, respectively, the singularity formation in free surfaces as a self-similar phenomenon in thin-film dynamics, the elastohydrodynamic lubrication by weakly viscoelastic fluids, and the inertia-free film flows under gravity with contact lines. It addresses graduate students and early-career researchers interested in theoretical fluid mechanics and its mathematical foundations, but also experienced scientists, actively employing perturbation analysis for long, who want to broaden their horizon.