A conformal mapping technique for solving two-dimensional nonlinear potential-flow problems involving free or interfacial boundaries is presented. These include free-surface waves, capillary waves, and flexural-gravity waves with complex bottom topography in the channel. When solving such problems, it is necessary at some stage to determine a function, such as the complex potential, the mapping function, or the complex velocity, from its values prescribed by the boundary conditions along the entire boundary of the flow domain. We briefly review the historical progress in solving nonlinear boundary-value problems based on the complex variable function theory. It starts from Kirchhoff conformal mapping technique and Joukowski and Michell hodograph method who introduced the concept of an auxiliary parameter plane, or ζ-plane. We also outline Chaplygin’s method of special points, which allows one to obtain a desired complex function without using the conformal-mapping technique in explicit form. This method consists of finding the required complex function from its zeros and poles in the flow region. Then, applying the Riemann–Schwarz symmetry principle and Liouville’s theorem, the desired function is obtained. We will show how Chaplygin’s special point technique can be applied to solve complex problems with the variable velocity magnitude on the free surface/ interface due to gravity, surface tension, or interaction with an elastic plate or another liquid. In other words, we will show how to derive integral formulae that determine the required complex function. We present case studies including flexural-gravity waves due to a body submerged in the uniform stream of infinite depth fluid [1] and in the channel of finite depth with arbitrary bottom topography [2]; surface tension effect on the quantisation of the waves in the hollow vortex in a corner geometry [3]; impulsive impact of a submerged body [4], etc. REFERENCES [1] Y.A. Semenov. Nonlinear flexural-gravity waves due to a body submerged in the uniform stream. Physics of Fluids, 33 (5), 052115, 2021. [2] C. Liang, B. Ni, Y.A. Semenov. Nonlinear flexural-gravity waves for flows over bottom topography. J. Fluid Mech. 1019, A40, 2025. [3] Y.A. Semenov, B. Ni. Quantization of free-surface waves in a hollow vortex with surface tension. J. Fluid Mech. 1026, A48, 2026. [4] Y.A. Semenov, Y.N. Savchenko, G.Y. Savchenko. Impulsive impact of a submerged body. J. of Fluid Mech., 919, R4, 2021.