The chapter introduces the field of fuzzy sets and fuzzy logic, which has evolved from its early days as an exotic research area to a rapidly growing field of interest in both engineering and academic communities. The term "fuzzy logic" has undergone several transformations, initially referring to tools from many-valued logic, then to fuzzy logic control, and now to a theory of approximate reasoning. The author, Siegfried Gottwald, traces his interest in this field back to his early work on set-theoretic foundations for fuzzy sets and highlights the influence of Polish control engineers, particularly Witold Pedrycz. The book presents a revised and extended collection of the author's essential results, focusing on fuzzy relation equations and fuzzy control. Each chapter is based on previous research and includes new material, with the final chapter combining results from various sources. The book also acknowledges the support of the SEL-Stiftung and the FG Intellektik, as well as the author's family for their support. The content covers logical preliminaries, basic fuzzy set theory, set equations with fuzzy sets, fuzzy controllers, and methodological issues, including comparisons of fuzzy sets, approximate solutions, and evaluation of fuzzy models.The chapter introduces the field of fuzzy sets and fuzzy logic, which has evolved from its early days as an exotic research area to a rapidly growing field of interest in both engineering and academic communities. The term "fuzzy logic" has undergone several transformations, initially referring to tools from many-valued logic, then to fuzzy logic control, and now to a theory of approximate reasoning. The author, Siegfried Gottwald, traces his interest in this field back to his early work on set-theoretic foundations for fuzzy sets and highlights the influence of Polish control engineers, particularly Witold Pedrycz. The book presents a revised and extended collection of the author's essential results, focusing on fuzzy relation equations and fuzzy control. Each chapter is based on previous research and includes new material, with the final chapter combining results from various sources. The book also acknowledges the support of the SEL-Stiftung and the FG Intellektik, as well as the author's family for their support. The content covers logical preliminaries, basic fuzzy set theory, set equations with fuzzy sets, fuzzy controllers, and methodological issues, including comparisons of fuzzy sets, approximate solutions, and evaluation of fuzzy models.