This book develops the fundamentals of Poisson geometry and deformation quantization in a coherent manner, starting from physical questions. Poisson geometry provides a general framework for geometric mechanics and represents a generalization of symplectic geometry. This plays an important role, particularly with regard to mechanical systems with symmetries and their phase space reduction. For the intended quantization, the geometric concepts are interpreted algebraically and formulated accordingly. Building on this, deformation quantization offers a general framework for the quantization of Poisson manifolds, which is presented here for the first time in textbook form. Central topics such as the Fedosov construction of star products and elements of the representation theory of deformed algebras are addressed.