Sampling has important applications in a wide range of areas, including molecular dynamics, Bayesian statistics and machine learning. Many of the most widely used dynamics for sampling have a kinetic structure, the two most prominent examples of which are underdamped Langevin dynamics and Hamiltonian Monte Carlo. We discuss the quantitative long-time L^2 convergence behavior of these kinetic dynamics for sampling, under different assumptions of the potential. For convex potentials, we show that kinetic sampling dynamics accelerate the convergence rate by a square root factor of the Poincaré constant, compared to the overdamped Langevin dynamics. We also show in the weakly confining setting, how the growth rate of the potential impacts the convergence rates in L^2 via weak variants of the Poincaré inequality. Finally, we will also discuss the recent progress in entropic convergence of these dynamics under convexity and log-Sobolev inequality.