Effective Evolution Equations from Quantum Dynamics These notes investigate the time evolution of quantum systems and in particular the rigorous derivation of effective equations approximating the many body Schrödinger dynamics in certain physically interesting regimes. The focus is primarily on the derivation of time dependent effective theories (non equilibrium question) approximating many body quantum dynamics. The book is divided into seven sections the first of which briefly reviews the main properties of many body quantum systems and their time evolution. Section 2 introduces the mean field regime for bosonic systems and explains how the many body dynamics can be approximated in this limit using the Hartree equation. Section 3 presents a method based on the use of coherent states for rigorously proving the convergence towards the Hartree dynamics while the fluctuations around the Hartree equation are considered in Section 4. Section 5 focuses on a discussion of a more subtle regime in which the many body evolution can be approximated by means of the nonlinear Gross Pitaevskii equation. Section 6 addresses fermionic systems (characterized by antisymmetric wave functions); here the fermionic mean field regime is naturally linked with a semiclassical regime and it is proven that the evolution of approximate Slater determinants can be approximated using the nonlinear Hartree Fock equation. In closing Section 7 reexamines the same fermionic mean field regime but with a focus on mixed quasi free initial data approximating thermal states at positive temperature. . Introduction Mean Field Regime for Bosonic Systems Coherent States Approach. Fluctuations Around Hartree Dynamics The Gross Pitaevskii Regime Mean Field regime for Fermionic Systems Dynamics of Fermionic Quasi Free Mixed States The Role of Correlations in the Gross Pitaevskii Energy.