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With applications in quantum field theory, elementary particle physics and general relativity, this two-volume work studies invariance of differential operators under Lie algebras, quantum groups, superalgebras including infinite-dimensional cases, Schroedinger algebras, applications to holography. This first volume covers the general aspects of Lie algebras and group theory supplemented by many concrete examples for a great variety of noncompact semisimple Lie algebras and groups. Contents:IntroductionLie Algebras and GroupsReal Semisimple Lie AlgebrasInvariant Differential OperatorsCase of the Anti-de Sitter GroupConformal Case in 4DKazhdan-Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant EquationsInvariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie AlgebrasMultilinear Invariant Differential Operators from New Generalized Verma ModulesBibliographyAuthor IndexSubject Index