This book provides essential foundations of mathematics, covering set theory, algebra, the theory of real and complex numbers, and topology. It serves as the basis for further exploration of mathematics. Not only are the necessary concepts introduced, but also essential – including deeper – statements are proven. The material is illustrated and supplemented by unusual examples and diverse exercises. The book is suitable for self-study, but primarily designed as companion reading from the beginning for a study in mathematics, physics, and computer science. Its coherent approach makes it highly readable and comparatively easy to understand. This book is a translation of the original German edition ‘Grundkonzepte der Mathematik’ by Uwe Storch and Hartmut Wiebe, published by Springer-Verlag GmbH, DE in 2017. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content (by the author Hartmut Wiebe only, as Uwe Storch is no longer with us).