After a short introduction to the program of deformation quantization we indicate why this program is of current interest. One of the reasons is Kontsevich’s generalization of the Moyal product of phase-space functions to the case of general Poisson manifolds. We discuss this generalization, including the graphical calculus for presenting the result. We then illustrate the techniques of deformation quantization for quantum mechanical problems by considering the case of the simple harmonic oscillator. We indicate the relations to more conventional approaches, including the formalisms involving operators in Hilbert space and path integrals. Finally, we sketch some new results for relativistic quantum field theories.