  A prevalent problem in general state-space models is the approximation of the
smoothing distribution of a state, or a sequence of states, conditional on the
observations from the past, the present, and the future. The aim of this paper
is to provide a rigorous foundation for the calculation, or approximation, of
such smoothed distributions, and to analyse in a common unifying framework
different schemes to reach this goal. Through a cohesive and generic exposition
of the scientific literature we offer several novel extensions allowing to
approximate joint smoothing distribution in the most general case with a cost
growing linearly with the number of particles.
