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After a short review of our understanding of how   a weak disorder or a low density of impurities affect the universality class of a phase transition,   I will discuss a simple tree-like toy model for which one can prove that,  due to   disorder, the depinning transition becomes  an infinite order  transition of the Berezinski-Kosterlitz-Thouless type.   This toy model was introduced  to understand the depinning transition of a line from a random substrate,  one of the simplest problems in the theory of disordered systems.   This depinning transition  has a long history among physicists and mathematicians.   Still  the nature of the transition, one of the central questions of the problem, remains  unsolved.