In this paper the concept of the stress intensity factor, the underlying mechanics problem leading to its emergence, and its physical relevance, particularly its relation to fracture mechanics are discussed, the reasons as to why it has become nearly an indispensable tool for studying such important phenomena as brittle fracture and fatigue or corrosion fatigue crack propagation in structural solids are considered, a brief discussion of some of the important methods of solution of elastic crack problems is given, and a number of related special mechanics problems are described.