In recent years, algorithmic graph theory has become increasingly important since it serves as a link between discrete mathematics and theoretical computer science. This textbook introduces students of mathematics and computer science to the interrelated fields of graph theory, algorithms and complexity. No specific previous knowledge is assumed. The central theme of the book is a geometrical problem dating back to Jakob Steiner. This problem, now called the Steiner tree problem, was initially of importance only within the context of land surveying. Recent applications, as diverse as VLSI-layout and the study of phylogenetic trees, have, however, lead to significant interest in the problem. The resulting progress has uncovered fascinating connections to and among graph theory, the study of algorithms and complexity. The single problem thus serves to bind and motivate these areas. The book’s topics include: exact algorithms; computational complexity; approximation algorithms; limits of approximability; randomness helps; the Manhattan Steiner problem; heuristics; packing of Steiner trees; and applications. A fundamental feature of the book is that each chapter ends with an excursion into some related area. These excursions reinforce the concepts and methods introduced for the Steiner tree problem by putting them in a broader context.