This book is about the foundations and applications of analytic hyperbolic geometry from the viewpoint of hyperbolic vectors, called gyrovectors. The underlying mathematical tools, gyrogroups and gyrovector spaces, are developed along analogies they share with groups and vector spaces. As a result, a gyrovector space approach to hyperbolic geometry, fully analogous to the standard vector space approach to Euclidean geometry, emerges. Owing to its strangeness, some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. But this book opens the door on its mission to make the hyperbolic geometry of Bolyai and Lobachevsky widely accessible by introducing a gyrovector space approach to hyperbolic geometry guided by analogies that it shares with the common vector space approach to Euclidean geometry. As a mathematical prerequisite for a fruitful reading of this book it is assumed familiarity with Euclidean geometry from the point of view of vectors and, occasionally, with differential calculus and functions of a complex variable. It includes both elementary and advanced topics, and is structured so that it can be enjoyed equally by undergraduates, graduate students, researchers and academics in geometry, algebra, mathematical physics, theoretical physics and astronomy.