Particle Filtering (PF) algorithms propagate in time a Monte Carlo (MC) approximation of the a posteriori filtering measure in a Hidden Markov Chain (HMC) model. In this paper we first shed some new light on two classical PF algorithms, which can be considered as natural MC implementations of two two-step direct recursive formulas for computing the filtering distribution. We next address the Particle Prediction (PP) problem, which happens to be simpler than the PF problem because the optimal prediction conditional importance distribution (CID) is much easier to sample from. Motivated by this result we finally develop two PP-based PF algorithms, and we compare our algorithms via simulations