In this talk, we shall design and analyze additive and multiplicative multilevel methods on adapted grids obtained by newest vertex bisection. The analysis relies on a novel decomposition of newest vertex bisection which give a bridge to transfer results on multilevel methods from uniform grids to adaptive grids. Based on this space decomposition, we will present a unified approach to the multilevel methods for $H^1$, $H(\rm curl)$, and $H(\rm div)$ systems.