This paper is the first part of a two-part work aiming to establish a unified theory of unitary representations of connected Lie groups. The author shows that by generalizing and rigorously defining the concept of quantizing a function, one can develop a theory that constructs many irreducible unitary representations of connected Lie groups. In the compact case, this theory includes the Borel-Weil theorem. Generalizing Kirillov's result for nilpotent groups, the author and Auslander show that this theory yields all irreducible unitary representations of solvable groups of type I, with a criterion for type I expressed in terms of the theory. For semi-simple groups, results by Harish-Chandra and Schmid suggest that this method constructs enough representations to decompose the regular representation. The theory is based on differential geometry. A key point is that the 2-form of a symplectic manifold, under certain conditions (integrity condition), is the curvature of a line bundle with connection. The Hilbert space involved is found among the sections of this line bundle, and operating on these sections forms a Lie algebra (under Poisson bracket) of functions on the manifold. This mapping of functions to operators is quantization. Polarizing the symplectic manifold is essential for extracting the Hilbert space and Lie algebra. This concept is broad enough to include classical quantum mechanical situations, such as the Bargmann-Segal-Fock representation of the Heisenberg Lie algebra. Part I focuses on pre-quantization. A representation of a group arises from a symplectic homogeneous space X when the corresponding Lie algebra of Hamiltonian vector fields can be lifted to quantizable functions. One result (Theorem 5.4.1) states that this is the case when X corresponds to or covers an orbit in the dual of the Lie algebra, justifying the idea of finding irreducible representations from these orbits. This also generalizes Wang's theorem for compact Kähler homogeneous spaces. Additionally, a corollary generalizes Borel-Weil results, showing that the 2-form on the orbit defined by a linear functional on the Lie algebra is integral if and only if the functional is the differential of a character on the isotropy group. Part I is devoted to the differential geometry foundations of the theory, including proofs of basic facts.This paper is the first part of a two-part work aiming to establish a unified theory of unitary representations of connected Lie groups. The author shows that by generalizing and rigorously defining the concept of quantizing a function, one can develop a theory that constructs many irreducible unitary representations of connected Lie groups. In the compact case, this theory includes the Borel-Weil theorem. Generalizing Kirillov's result for nilpotent groups, the author and Auslander show that this theory yields all irreducible unitary representations of solvable groups of type I, with a criterion for type I expressed in terms of the theory. For semi-simple groups, results by Harish-Chandra and Schmid suggest that this method constructs enough representations to decompose the regular representation. The theory is based on differential geometry. A key point is that the 2-form of a symplectic manifold, under certain conditions (integrity condition), is the curvature of a line bundle with connection. The Hilbert space involved is found among the sections of this line bundle, and operating on these sections forms a Lie algebra (under Poisson bracket) of functions on the manifold. This mapping of functions to operators is quantization. Polarizing the symplectic manifold is essential for extracting the Hilbert space and Lie algebra. This concept is broad enough to include classical quantum mechanical situations, such as the Bargmann-Segal-Fock representation of the Heisenberg Lie algebra. Part I focuses on pre-quantization. A representation of a group arises from a symplectic homogeneous space X when the corresponding Lie algebra of Hamiltonian vector fields can be lifted to quantizable functions. One result (Theorem 5.4.1) states that this is the case when X corresponds to or covers an orbit in the dual of the Lie algebra, justifying the idea of finding irreducible representations from these orbits. This also generalizes Wang's theorem for compact Kähler homogeneous spaces. Additionally, a corollary generalizes Borel-Weil results, showing that the 2-form on the orbit defined by a linear functional on the Lie algebra is integral if and only if the functional is the differential of a character on the isotropy group. Part I is devoted to the differential geometry foundations of the theory, including proofs of basic facts.