To describe dynamics of molecules in classical mechanics, it is necessary to solve Newton’s equations. On a quantum level, the corresponding equation is the Schrödinger equation of many-body quantum dynamics. To develop corresponding numerical methods for the latter poses a difficult problem for numerical analysis. This book contains a description of numerical methods for some intermediate models between the classical and quantum cases. The first chapter contains a short review of the necessary notions from quantum mechanics. In chapter 2, time-dependent variational principles are used for a construction of basic intermediate models. Numerical methods for the linear time-dependent Schrödinger equation (in moderate dimensions) are introduced in chapter 3 and the case of nonlinear reduced models is treated in chapter 4. Hagedorn wave packets are used for the time-dependent Schrödinger equation in a semi-classical scaling in the last chapter. The book illustrates the usefulness of mutual interplay between numerical analysis and its applications in mathematical physics.