Hamiltonian Monte Carlo (HMC) has proven highly effective in solving complex statistical problems, but its theoretical underpinnings have been largely confined to differential geometry, making it difficult for applied statisticians to understand and implement. This review aims to provide a conceptual introduction to HMC, focusing on developing a principled intuition rather than rigorous mathematical details. The author, Michael Betancourt, a research scientist at Columbia University, emphasizes the importance of understanding how HMC works, when it succeeds, and when it fails. The review begins by discussing the challenges of computing expectations in high-dimensional spaces, where the geometry of probability distributions can frustrate efficient statistical computing. It then introduces Markov Chain Monte Carlo (MCMC) from a geometric perspective, highlighting the need for effective methods to explore the typical set, a region of high probability mass that is crucial for accurate expectation estimation. The core of the review is the introduction of HMC, which leverages the geometry of phase space and Hamilton's equations to generate coherent trajectories through the typical set. By introducing auxiliary momentum parameters, HMC ensures conservative dynamics, preserving volume in phase space and enabling efficient exploration. The review also covers practical aspects of implementing HMC, including symplectic integrators, tuning, and diagnostics for pathological behavior. Overall, the review provides a comprehensive conceptual account of HMC's theoretical foundations, making it accessible to practitioners and statisticians, while also highlighting the importance of further exploration and development in this area.Hamiltonian Monte Carlo (HMC) has proven highly effective in solving complex statistical problems, but its theoretical underpinnings have been largely confined to differential geometry, making it difficult for applied statisticians to understand and implement. This review aims to provide a conceptual introduction to HMC, focusing on developing a principled intuition rather than rigorous mathematical details. The author, Michael Betancourt, a research scientist at Columbia University, emphasizes the importance of understanding how HMC works, when it succeeds, and when it fails. The review begins by discussing the challenges of computing expectations in high-dimensional spaces, where the geometry of probability distributions can frustrate efficient statistical computing. It then introduces Markov Chain Monte Carlo (MCMC) from a geometric perspective, highlighting the need for effective methods to explore the typical set, a region of high probability mass that is crucial for accurate expectation estimation. The core of the review is the introduction of HMC, which leverages the geometry of phase space and Hamilton's equations to generate coherent trajectories through the typical set. By introducing auxiliary momentum parameters, HMC ensures conservative dynamics, preserving volume in phase space and enabling efficient exploration. The review also covers practical aspects of implementing HMC, including symplectic integrators, tuning, and diagnostics for pathological behavior. Overall, the review provides a comprehensive conceptual account of HMC's theoretical foundations, making it accessible to practitioners and statisticians, while also highlighting the importance of further exploration and development in this area.