DescriptionThe dissertation is devoted to the applications of the Noncommutative Geometry Program to the study of Integrable Systems and Cluster Algebras. In particular, it is shown that cluster algebras introduced by A.~Goncharov and R.~Kenyon admit a noncommutative generalization. This generalization can be viewed as a family of categories equipped with a double Quasi Poisson bracket and a family of functors between these categories which preserve the double bracket. From this perspective, the commutative cluster algebra appears as the coordinate ring of the moduli space of one dimensional representations of the noncommutative cluster algebra. It is shown that Noncommutative systems of ODEs, suggested earlier by M.~Kontsevich and A.~Usnich, admit a formulation as Noncommutative Hamilton flows. Finally, a non-skew-symmetric generalization of the double Poisson bracket is considered. It is shown that such modified double Poisson brackets inherit major properties of double Poisson brackets.