The universe is pervaded by particles with extreme energies, millions of times greater than we can produce on Earth. They have been a mystery for over a century. Now, current and future experiments in particle astrophysics are leading us to answers to the most fundamental questions about them. How does nature accelerate the highest energy particles in the universe? Do new interactions between them occur at such extreme energies? Are there unknown aspects of spacetime that can be uncovered by studying these particles? This book brings together three fields within "extreme astronomy": ultra-high-energy cosmic ray physics, neutrino astronomy, and gamma-ray astronomy, and discusses how each can help answer these questions. Each field is presented with a theoretical introduction that clearly elucidates the key questions scientists face. This is followed by chapters that discuss the current set of experiments - how they work and their discoveries. Finally, new techniques and approaches are discussed to solve the mysteries uncovered by the current experiments.