This paper presents the theory of differential calculus on compact matrix pseudogroups, which are generalizations of Lie groups known as quantum groups. The author develops the general theory of differential calculus on quantum groups, focusing on bicovariant bimodules, which are analogous to tensor bundles over Lie groups. The paper describes tensor algebra and exterior algebra constructions and shows that any bicovariant first-order differential calculus can be naturally lifted to the exterior algebra, allowing for the definition of higher-order exterior derivatives. The Cartan-Maurer formula is derived, and a bilinear operation on the dual space of left-invariant differential forms is introduced, playing the role of the Lie bracket. Generalized antisymmetry and Jacobi identities are also proved. The paper emphasizes the importance of differential geometry in Lie group theory and suggests a similar interplay between differential geometry and group theory in non-commutative geometry, where quantum groups replace Lie groups. The main aim is to present differential calculus on quantum groups, showing that classical Lie group concepts can be generalized. The paper uses an axiomatic approach to define first-order differential calculus, which includes properties of the exterior derivative and left- and right-covariance. It is noted that for non-commutative geometry, a functional definition of differential calculus is not possible, and once the first-order calculus is fixed, further notions are introduced functorially. Section 1 introduces the main definitions and results of first-order differential calculus, showing that reasonable calculi correspond to right ideals in the algebra of smooth functions on quantum groups. Section 2 discusses bicovariant bimodules, which correspond to vector bundles over group manifolds with left and right group actions. It proves that such bundles are topologically trivial and introduces the adjoint representation. Section 3 focuses on the twisted flip automorphism associated with bicovariant bimodules, which satisfies the braid equation and allows for the introduction of antisymmetrization operations on tensor powers of the bimodule.This paper presents the theory of differential calculus on compact matrix pseudogroups, which are generalizations of Lie groups known as quantum groups. The author develops the general theory of differential calculus on quantum groups, focusing on bicovariant bimodules, which are analogous to tensor bundles over Lie groups. The paper describes tensor algebra and exterior algebra constructions and shows that any bicovariant first-order differential calculus can be naturally lifted to the exterior algebra, allowing for the definition of higher-order exterior derivatives. The Cartan-Maurer formula is derived, and a bilinear operation on the dual space of left-invariant differential forms is introduced, playing the role of the Lie bracket. Generalized antisymmetry and Jacobi identities are also proved. The paper emphasizes the importance of differential geometry in Lie group theory and suggests a similar interplay between differential geometry and group theory in non-commutative geometry, where quantum groups replace Lie groups. The main aim is to present differential calculus on quantum groups, showing that classical Lie group concepts can be generalized. The paper uses an axiomatic approach to define first-order differential calculus, which includes properties of the exterior derivative and left- and right-covariance. It is noted that for non-commutative geometry, a functional definition of differential calculus is not possible, and once the first-order calculus is fixed, further notions are introduced functorially. Section 1 introduces the main definitions and results of first-order differential calculus, showing that reasonable calculi correspond to right ideals in the algebra of smooth functions on quantum groups. Section 2 discusses bicovariant bimodules, which correspond to vector bundles over group manifolds with left and right group actions. It proves that such bundles are topologically trivial and introduces the adjoint representation. Section 3 focuses on the twisted flip automorphism associated with bicovariant bimodules, which satisfies the braid equation and allows for the introduction of antisymmetrization operations on tensor powers of the bimodule.