Hochschild Cohomology is a mathematical concept used in the field of algebra, particularly in the study of associative algebras. It provides a way to measure the "deformations" of algebras and their modules, capturing information about their structure and relationships. This cohomology theory is named after Gerhard Hochschild, who introduced it in the 1940s. The cohomology groups are constructed using chain complexes and involve the use of multilinear maps. These groups can reveal important properties of algebras, such as their homological dimensions and the existence of certain types of extensions. Hochschild cohomology has applications in various areas, including representation theory and noncommutative geometry.