Hamiltonian Monte Carlo (HMC) has become a powerful tool in statistical computing, but its theoretical foundations are rooted in differential geometry, which limits its accessibility to applied statisticians. This review provides a conceptual introduction to HMC, focusing on its principles and optimal implementation rather than rigorous mathematical details. The goal is to convey the intuition behind HMC's success in high-dimensional problems. HMC is based on the idea of exploring the typical set of a probability distribution, which is the region where most of the probability mass lies. In high-dimensional spaces, the volume of the typical set is concentrated away from the mode, making it crucial to efficiently explore this region. Markov chain Monte Carlo (MCMC) methods, including HMC, are used to sample from complex distributions by constructing Markov transitions that preserve the target distribution. The Metropolis-Hastings algorithm is a common MCMC method that proposes new states and accepts or rejects them based on the target distribution. However, it can be inefficient in high dimensions due to the geometry of the target distribution. HMC addresses this by incorporating information about the geometry of the target distribution through the use of Hamiltonian dynamics. Hamiltonian dynamics involve a phase space consisting of both position (parameters of the target distribution) and momentum (auxiliary variables). The Hamiltonian function, which is the sum of kinetic and potential energy, defines the dynamics of the system. By following the Hamiltonian equations of motion, HMC can efficiently explore the typical set of the target distribution. The idealized Hamiltonian Markov transition involves generating trajectories in phase space that preserve the volume of the phase space. These trajectories are then projected back onto the target parameter space to generate samples. The key to HMC's efficiency is its ability to move through the typical set in a coherent manner, avoiding the diffusive behavior of other MCMC methods. The practical implementation of HMC involves choosing appropriate kinetic energy functions and integration times to ensure efficient sampling. Symplectic integrators are used to approximate the Hamiltonian dynamics, and careful tuning is required to balance the trade-off between exploration and computational cost. In conclusion, HMC provides a principled approach to sampling from complex distributions by leveraging the geometry of the target distribution. Its efficiency in high-dimensional spaces makes it a valuable tool in applied statistics, although its theoretical foundations are rooted in differential geometry, which can limit its accessibility to some practitioners.Hamiltonian Monte Carlo (HMC) has become a powerful tool in statistical computing, but its theoretical foundations are rooted in differential geometry, which limits its accessibility to applied statisticians. This review provides a conceptual introduction to HMC, focusing on its principles and optimal implementation rather than rigorous mathematical details. The goal is to convey the intuition behind HMC's success in high-dimensional problems. HMC is based on the idea of exploring the typical set of a probability distribution, which is the region where most of the probability mass lies. In high-dimensional spaces, the volume of the typical set is concentrated away from the mode, making it crucial to efficiently explore this region. Markov chain Monte Carlo (MCMC) methods, including HMC, are used to sample from complex distributions by constructing Markov transitions that preserve the target distribution. The Metropolis-Hastings algorithm is a common MCMC method that proposes new states and accepts or rejects them based on the target distribution. However, it can be inefficient in high dimensions due to the geometry of the target distribution. HMC addresses this by incorporating information about the geometry of the target distribution through the use of Hamiltonian dynamics. Hamiltonian dynamics involve a phase space consisting of both position (parameters of the target distribution) and momentum (auxiliary variables). The Hamiltonian function, which is the sum of kinetic and potential energy, defines the dynamics of the system. By following the Hamiltonian equations of motion, HMC can efficiently explore the typical set of the target distribution. The idealized Hamiltonian Markov transition involves generating trajectories in phase space that preserve the volume of the phase space. These trajectories are then projected back onto the target parameter space to generate samples. The key to HMC's efficiency is its ability to move through the typical set in a coherent manner, avoiding the diffusive behavior of other MCMC methods. The practical implementation of HMC involves choosing appropriate kinetic energy functions and integration times to ensure efficient sampling. Symplectic integrators are used to approximate the Hamiltonian dynamics, and careful tuning is required to balance the trade-off between exploration and computational cost. In conclusion, HMC provides a principled approach to sampling from complex distributions by leveraging the geometry of the target distribution. Its efficiency in high-dimensional spaces makes it a valuable tool in applied statistics, although its theoretical foundations are rooted in differential geometry, which can limit its accessibility to some practitioners.