Textbook in PDF format Fatigue failure of engineering materials and structures has long been a great challenge for structural integrity, reliability and safety in mechanical, civil and aerospace engineering. These failure mechanisms and their modeling are critical concerns for managing aging structures, and directly affect sustainability across society. In this context, the fundamental theories and methods of fatigue failure of engineering materials and structures are discussed in detail. Fatigue damage accumulation, crack initiation and crack growth analysis are presented from materials to structures, deterministic to probabilistic fatigue, physics to data science, uniaxial to multiaxial fatigue, and extremely low cycle fatigue to very high cycle fatigue. The focus is on mechanical understanding and risk management for design, maintenance, and operation