This groundbreaking monograph is the first to focus on the classification of hyperbolic root systems, which are crucial to the theory of Lorentzian (or hyperbolic) Kac-Moody algebras. The author presents finite results for these hyperbolic root systems, delving into the classification for the first non-trivial and most complex case of rank three. This endeavor involves extensive and intricate calculations. This work can be seen as the foundation for developing a comprehensive theory of Lorentzian Kac-Moody algebras in the rank three case, which serves as the hyperbolic counterpart to sl2. Targeted at researchers, graduate students, and undergraduates interested in the theory of algebras and Lie groups, algebraic geometry, as well as mathematical and theoretical physics, this monograph opens new avenues for exploration in these fields.