This course will cover the introduction of Tate-Hochschild homology and cohomology of modules with examples. Also it will cover the introduction and the survey of Fomin and Zelevinsky’s theory of cluster algebras. It is one of the most important recent developments of algebraic combinatorics. Cluster algebras are a class of combinatorially defined commutative rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts.