  Computing smoothing distributions, the distributions of one or more states
conditional on past, present, and future observations is a recurring problem
when operating on general hidden Markov models. The aim of this paper is to
provide a foundation of particle-based approximation of such distributions and
to analyze, in a common unifying framework, different schemes producing such
approximations. In this setting, general convergence results, including
exponential deviation inequalities and central limit theorems, are established.
In particular, time uniform bounds on the marginal smoothing error are obtained
under appropriate mixing conditions on the transition kernel of the latent
chain. In addition, we propose an algorithm approximating the joint smoothing
distribution at a cost that grows only linearly with the number of particles.
