X-ray absorption fine structure spectroscopy (XAFS) is a unique probe for local atomic and electronic structure of absorbing centers in particulate science, physics, chemistry, and biology. During the past several decades, XAFS has gained dramatic advances in every aspect, including theoretical explanation and experimental applications. In this review, we outline the basic physics underlying the X-ray absorption process, experimental design, data analysis for both extended X-ray absorption fine structure (EXAFS), which is one of the mainstream directions of absorption spectroscopy and allows high-accuracy determination of the parameters of the short-range order in multicomponent amorphous and quasi-crystalline substances, and X-ray absorption near-edge structure (XANES). The modern advances and future outlook are also briefly included.