This paper proves that any finite-dimensional Poisson manifold can be canonically quantized, meaning that the set of equivalence classes of associative algebras close to function algebras on manifolds is in one-to-one correspondence with the set of equivalence classes of Poisson manifolds modulo diffeomorphisms. This result is a corollary of the author's "Formality conjecture," which was proposed around 1993-1994. The solution presented here uses ideas from string theory, and the formulas can be viewed as a perturbation series for a topological two-dimensional quantum field theory coupled with gravity. The paper is structured into several sections: 1. An elementary introduction to deformation quantization and the main statement. 2. An explicit formula for deformation quantization in coordinates. 3. An introduction to deformation theory using differential graded Lie algebras. 4. A geometric reformulation of the theory using odd vector fields on formal supermanifolds, introducing notions like $L_{\infty}$-morphisms and quasi-isomorphisms. 5. Tools for constructing the quasi-isomorphism, including compactified configuration spaces and differential polynomials. 6. Proof that the machinery establishes the Formality conjecture for affine spaces. 7. Extension of the results to general manifolds using formal geometry and superconnections. 8. Additional structures in deformation theory, such as the cup-product on the supermoduli space, and applications to representation theory and algebraic geometry. The author plans to write a complement to the paper, covering topics such as comparisons with other star-product constructions, reformulations of the Formality conjecture, arithmetic properties of coefficients, applications to mirror symmetry, and a Lagrangian for a quantum field theory.This paper proves that any finite-dimensional Poisson manifold can be canonically quantized, meaning that the set of equivalence classes of associative algebras close to function algebras on manifolds is in one-to-one correspondence with the set of equivalence classes of Poisson manifolds modulo diffeomorphisms. This result is a corollary of the author's "Formality conjecture," which was proposed around 1993-1994. The solution presented here uses ideas from string theory, and the formulas can be viewed as a perturbation series for a topological two-dimensional quantum field theory coupled with gravity. The paper is structured into several sections: 1. An elementary introduction to deformation quantization and the main statement. 2. An explicit formula for deformation quantization in coordinates. 3. An introduction to deformation theory using differential graded Lie algebras. 4. A geometric reformulation of the theory using odd vector fields on formal supermanifolds, introducing notions like $L_{\infty}$-morphisms and quasi-isomorphisms. 5. Tools for constructing the quasi-isomorphism, including compactified configuration spaces and differential polynomials. 6. Proof that the machinery establishes the Formality conjecture for affine spaces. 7. Extension of the results to general manifolds using formal geometry and superconnections. 8. Additional structures in deformation theory, such as the cup-product on the supermoduli space, and applications to representation theory and algebraic geometry. The author plans to write a complement to the paper, covering topics such as comparisons with other star-product constructions, reformulations of the Formality conjecture, arithmetic properties of coefficients, applications to mirror symmetry, and a Lagrangian for a quantum field theory.