The two-state vector formalism of quantum mechanics is a time-symmetrized approach to standard quantum theory particularly helpful for the analysis of experiments performed on pre-and post-selected ensembles. It allows to see numerous peculiar effects which naturally arise in this approach. In particular, the concepts of “weak measurements”(standard measurements with weakening of the interaction) and “weak values”(the outcomes of weak measurements) reveal a very unusual but consistent picture. Recently, more and more effects are viewed as manifestations of weak measurements and more and more weak measurement experiments have been performed. The polemic about the validity of the approach and the meaning of its concepts never stopped. The number of papers written on the subject almost doubled since publication of the first version of the review. The current review does not explain in details the new results, but it puts the development of the approach in the proper context and provides citations for further reading.