Nonlocal magnetization dynamics in ferromagnetic heterostructures Yaroslav Tserkovnyak, Arne Brataas, Gerrit E. W. Bauer, and Bertrand I. Halperin Two effects modify the GHz magnetization dynamics of nanoscale heterostructures of ferromagnetic and normal materials compared to isolated magnetic constituents: spin angular-momentum flow from a time-dependent ferromagnetic magnetization into adjacent materials, and spin angular momentum transfer between ferromagnets by an applied bias, causing mutual torques on the magnetizations. These phenomena are nonlocal, governed by the entire spin-coherent region limited by spin-flip relaxation processes. The review discusses recent progress in understanding magnetization dynamics in ferromagnetic heterostructures from first principles, focusing on spin pumping in layered structures. The main theory is semiclassical, based on a mean-field Stoner or spin-density-functional picture, but quantum-size effects and electron-electron correlations are also discussed. Many experiments support the theoretical predictions. The formalism is useful for understanding the physics and engineering characteristics of small devices like magnetic random-access memory elements. The review covers developments in understanding magnetization dynamics in ferromagnetic and normal conductor heterostructures over the last five years. It is timely as much of the basic physics is well understood. A consistent and coherent picture has evolved based on the diffusion equation for bulk transport in metallic ferromagnets and normal conductors with quantum-mechanical boundary conditions at interfaces. Noncollinearity of magnetization directions in structures with more than one magnet is essential for understanding the physics. The review focuses on dynamic effects, referring to a separate article for static transport properties. The present review extends static magnetoelectronic circuit theory to time-dependent phenomena, providing a framework to include two physical effects: spin-transfer torque from applied currents and spin pumping by moving ferromagnets into adjacent conductors. The theory is derived from microscopic principles, with material-dependent input parameters accessible to ab initio calculations. The review focuses on quasi-one-dimensional models, such as layered pillar structures. It does not attempt accurate modeling of concrete device structures, but the theory can be generalized to treat such situations. The review focuses on adiabatic effects to lowest order in the characteristic Larmor frequency. Despite limitations, the agreement with various experiments is gratifying. Effects beyond the model may cause observable phenomena. For example, quantum interference leading to inversion of magnetoresistance in high-quality tunnel junctions cannot be treated by the diffusion equation. Nonlinearities require numerical simulations or stability analysis based on dynamic systems theory. High temperatures and currents can be treated by stochastic methods, but input parameters are provided here. Current-induced dynamics of domain walls are also beyond the macrospin considerations of this review. The review discusses nonlocal exchange coupling and giant magnetoresistance, which are essential for understanding the physics. The discovery that the energy of magnetic multilayers depends on the relative direction of individualNonlocal magnetization dynamics in ferromagnetic heterostructures Yaroslav Tserkovnyak, Arne Brataas, Gerrit E. W. Bauer, and Bertrand I. Halperin Two effects modify the GHz magnetization dynamics of nanoscale heterostructures of ferromagnetic and normal materials compared to isolated magnetic constituents: spin angular-momentum flow from a time-dependent ferromagnetic magnetization into adjacent materials, and spin angular momentum transfer between ferromagnets by an applied bias, causing mutual torques on the magnetizations. These phenomena are nonlocal, governed by the entire spin-coherent region limited by spin-flip relaxation processes. The review discusses recent progress in understanding magnetization dynamics in ferromagnetic heterostructures from first principles, focusing on spin pumping in layered structures. The main theory is semiclassical, based on a mean-field Stoner or spin-density-functional picture, but quantum-size effects and electron-electron correlations are also discussed. Many experiments support the theoretical predictions. The formalism is useful for understanding the physics and engineering characteristics of small devices like magnetic random-access memory elements. The review covers developments in understanding magnetization dynamics in ferromagnetic and normal conductor heterostructures over the last five years. It is timely as much of the basic physics is well understood. A consistent and coherent picture has evolved based on the diffusion equation for bulk transport in metallic ferromagnets and normal conductors with quantum-mechanical boundary conditions at interfaces. Noncollinearity of magnetization directions in structures with more than one magnet is essential for understanding the physics. The review focuses on dynamic effects, referring to a separate article for static transport properties. The present review extends static magnetoelectronic circuit theory to time-dependent phenomena, providing a framework to include two physical effects: spin-transfer torque from applied currents and spin pumping by moving ferromagnets into adjacent conductors. The theory is derived from microscopic principles, with material-dependent input parameters accessible to ab initio calculations. The review focuses on quasi-one-dimensional models, such as layered pillar structures. It does not attempt accurate modeling of concrete device structures, but the theory can be generalized to treat such situations. The review focuses on adiabatic effects to lowest order in the characteristic Larmor frequency. Despite limitations, the agreement with various experiments is gratifying. Effects beyond the model may cause observable phenomena. For example, quantum interference leading to inversion of magnetoresistance in high-quality tunnel junctions cannot be treated by the diffusion equation. Nonlinearities require numerical simulations or stability analysis based on dynamic systems theory. High temperatures and currents can be treated by stochastic methods, but input parameters are provided here. Current-induced dynamics of domain walls are also beyond the macrospin considerations of this review. The review discusses nonlocal exchange coupling and giant magnetoresistance, which are essential for understanding the physics. The discovery that the energy of magnetic multilayers depends on the relative direction of individual