In this paper we present the two-state vector formalism of quantum mechanics. It is a time-symmetrized approach to standard quantum theory particularly helpful for the analysis of experiments performed on pre-and post-selected ensembles. Several peculiar effects which naturally arise in this approach are considered. In particular, the concept of “weak measurements’’ (standard measurements with weakening of the interaction) is discussed in depth revealing a very unusual but consistent picture. Also, a design of a gedanken experiment which implements a kind of quantum “time machine’’ is described. The issue of time-symmetry in the context of the two-state vector formalism is clarified.