For a class of semilinear wave equations we developed a technique which provides pointwise estimates of the solutions. It offers a qualitative as well as a quantitative asymptotic information on the solutions. It also provides a link between the nonlinear terms present in the equation and the asymptotic effects they generate. After introducing the technique I will discuss some nonlinear phenomena which persist even in the weak field regime but are absent in the linear approximation: the nonlinear late time tails and the nonlinear distortion of the traveling waves. These effects can be important for the gravitational waves being the weak field solutions of the Einstein equations.