  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.
