Hamiltonian Monte Carlo is a very popular MCMC method amongst Bayesian statisticians to get samples from a posterior distribution. This algorithm relies on the discretization of Hamiltonian dynamics which leave the target density invariant combined with a Metropolis step. In this talk, we will discuss convergence properties of this method to sample from a positive target density p on $R^d$ with either a fixed or a random numbers of integration steps. More precisely, we will present some mild conditions on p to ensure φ-irreducibility and ergodicity of the associated chain. We will also present verifiable conditions which imply geometric convergence. We will conclude with the introduction of new exact continuous time MCMC methods, and in particular the Bouncy Particle Sampler for which new theoretical results will be given.