The present thesis deals with the problems of simulation from a given target distribution and the estimation of ratios of normalizing constants, i.e. marginal likelihoods (ML). Both problems could be considerably difficult even for the simplest possible real-world statistical setups. We investigate how the combination of Hamiltonian Monte Carlo (HMC) and Sequential Monte Carlo (SMC) could be used to sample effectively from a multi-modal target distribution and to estimate ratios of normalizing constants at the same time. We call this novel combination Hamiltonian SMC (HSMC) algorithm and we show that it achieves some improvements over the existing Monte Carlo sampling algorithms, especially when the target distribution is multi-modal and/ or have complicated covariance structure. An important convergence result is proved for the HSMC, as well as an upper bound on the bias of the estimate of the ratio of MLs. Our investigation of the continuous time limit of the HSMC process reveals an interesting connection between Monte Carlo simulation and physics. We also concern ourselves with the problem of estimation of the uncertainty of the estimate of the ML of a HMM. We propose a new algorithm (Pairs algorithm) to estimate the non-asymptotic second moment of the estimate of the ML for general HMM. Later we show that there exists a linear-in-time bound on the relative variance of the estimate of the second moment of the ML obtained using the Pairs algorithm. This theoretical property of the relative variance translates in practice into a more reliable estimates of the second moment of the estimate of the MLs compared to the standard approach of running independent copies of the particle filter. We support out investigations with different numerical examples like Bayesian inference of a heteroscedastic regression, inference of a Lotka - Volterra based HMM, etc.