This Open access book provides an overview of the cutting-edge research to systematize innovative foundations in algorithmic theories and techniques, and apply them to solve socially significant real-life problems, thereby driving social transformation and advancement. Formulating such social problems into well-defined mathematical or computational terms requires not only a solid theoretical foundation but also a deep understanding of the application domain itself. The content is divided into three parts. Following an introductory chapter in Part I, Part II discusses how to bridge the gap between algorithmic solutions and social advancement. Chapters 2, 3, and 4 present motivating problems that showcase the remarkable performance improvements driven by algorithmic technologies, followed by examples of how interfaces and tools were provided to apply these solutions to real-world challenges. The subsequent Chapters 5 through 12 then focus on interdisciplinary discussions regarding how to formulate real societal issues into a catalog of problems to be addressed. Finally, Part III, spanning Chapters 13 to 23, highlights selected topics on the innovative algorithmic foundations developed through the research project, focusing on intriguing subjects and the latest ongoing research. These include processing large-scale discrete structures, graph algorithms, discrete optimization, quantum algorithms, and various other topics related to algorithmic foundations. This book will serve as a valuable resource for readers interested in the use of algorithms in real-life problems. This book is based on the research project in Japan, Algorithmic Foundations for Social Advancement.