This book discusses the intricate realms of geometric function theory and nonlinear potential theory. It offers a distinctive and integrative perspective rooted in the geometric and topological properties of mappings. By synthesizing these foundational concepts, it presents a fresh and accessible approach tailored to researchers and graduate students alike. Key topics include nonsmooth calculus, boundary behavior of functions and the theory of quasiconformal and quasiregular mappings, with practical applications to open-ended analytical problems. The methodologies span a broad spectrum—from real and complex analysis to classical potential theory—underscoring their versatility across mathematical disciplines.

The book establishes a solid foundation for readers eager to pursue independent research in these rapidly evolving fields. Through thoughtfully structured exercises, it offers a rich and engaging learning experience. A carefully curated bibliography supports further exploration, bridging classical foundations with emerging directions in contemporary research. Whether used in classrooms or for self-study, this volume serves as an indispensable resource for those seeking to deepen their understanding and actively contribute to the advancement of mathematical analysis.

Olli Martio is Professor in the Department of Mathematics and Statistics, University of Helsinki, Finland. With more than 200 papers and 3 books published, including Nonlinear Potential Theory of Degenerate Elliptic Equations, he is one of the pioneers in fields such as quasiconformal mappings and non-linear potential theory. He is the founder of the Finnish School of Nonlinear Potential Theory. Many of his mathematical descendants hold professor positions in Finland and other countries. He has acted as Editor of several journals, including Acta Mathematica.

Matti Vuorinen is Professor in the Department of Mathematics and Statistics, University of Turku, Finland. With more than 200 papers and 3 books, including Conformally Invariant Metrics and Quasiconformal Mappings, he is one of the pioneers in fields such as in quasiconformal mappings and geometric function theory. He has supervised 15 Ph.D. students and has hosted more than 100 research visits to Finland. His network of coauthors contains 100 mathematicians from the USA, Japan, China, India, Russia and European countries.