  This paper starts with a brief review of the topic of strong and weak pre-
and post-selected (PPS) quantum measurements, as well as weak values, and
afterwards presents original work. In particular, we develop a nonperturbative
theory of weak PPS measurements of an arbitrary system with an arbitrary meter,
for arbitrary initial states. New and simple analytical formulas are obtained
for the average and the distribution of the meter pointer variable, which hold
to all orders in the weak value. In the case of a mixed preselected state, in
addition to the standard weak value, an associated weak value is required to
describe weak PPS measurements. In the linear regime, the theory provides the
generalized Aharonov-Albert-Vaidman formula. Moreover, we reveal two new
regimes of weak PPS measurements: the strongly-nonlinear regime and the
inverted region, where the system-dependent contribution to the pointer
deflection decreases with increasing the measurement strength. The optimal
conditions for weak PPS measurements are achieved in the strongly-nonlinear
regime, where the magnitude of the average pointer deflection is equal or close
to the maximum. This maximum is independent of the measurement strength, being
typically of the order of the pointer uncertainty. We show that the
amplification in the weak PPS measurements is a product of two qualitatively
different quantities: proper amplification and enhancement. The effects of the
free system and meter Hamiltonians are discussed. We also identify optimal
meters for weak measurements. Exact solutions are obtained for a certain class
of the measured observables. These solutions are used for numerical
calculations, the results of which agree with the theory. Moreover, the theory
is extended to allow for a completely general post-selection measurement. We
also discuss time-symmetry properties of PPS measurements of any strength.
