In the last forty years, the study of singularity formation has mostly concerned model problems and focusing non linearities. In this second lecture, we will try to give a unified overview on the known mechanisms of singularity formation, with in particular the description of type I (self similar) versus type II (non self similar) singularities. We will present both construction and sometimes complete classification theorems, and will emphasize the key role played by specific non linear bubbles of energy: solitons.