Magnetohydrodynamics (MHD) studies the dynamics of magnetic fields in electrically conducting fluids. In addition to the sound wave and electromagnetic wave behaviors, magneto-fluids also exhibit an interesting phenomenon: They can produce the Alfven waves, which were first described in a physics paper by Hannes Alfven in 1942. Subsequently, Alfven was awarded the Nobel prize for his fundamental work on MHD with fruitful applications in plasma physics, in particular the discovery of Alfven waves. In this talk, we study (and construct) global solutions for the three dimensional incompressible MHD systems (with or without viscosity) in strong magnetic backgrounds. We present a complete and self-contained mathematical proof of the global nonlinear stability of Alfven waves. This is a joint work with Li Xu(AMSS) and Pin Yu(Tsinghua University).