This chapter consists of three main sections on the topics of nonlinear smoothing, the high-low decomposition method, and the method of almost conserved quantities. The nonlinear smoothing is the property that the difference of the solution and the linear solution with the same initial data is smoother than the initial data. We establish this property for the Korteweg-deVries and the cubic nonlinear Schrodinger equations. The high-low decomposition method is a useful tool in establishing global wellposedness for infinite energy data by splitting the initial data in low and high frequencies. This method relies heavily on smoothing estimates. Finally the method of almost conservation laws is a refinement of the high-low method that allows one to establish global wellposedness in even lower regularity spaces.