The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo (MALA and HMC) methods defined on Riemann manifolds to address the limitations of traditional MCMC algorithms when sampling from high-dimensional, strongly correlated target densities. These methods automatically adapt to the local structure of the parameter space, improving efficiency and convergence. The Riemann manifold framework allows for more efficient proposal mechanisms by incorporating the geometry of the target density, reducing the need for manual tuning of proposal densities. The methods are evaluated on logistic regression, log-Gaussian Cox point processes, stochastic volatility models, and Bayesian dynamic systems. The results show significant improvements in time-normalized effective sample size compared to alternative methods. MATLAB code is provided for replication. Key concepts include the Fisher-Rao metric tensor, geometric concepts in MCMC, and the use of Riemannian geometry to define the metric tensor. The paper demonstrates how Riemann manifold MALA (MMALA) and Riemann manifold HMC (RMHMC) methods can be implemented to improve sampling efficiency in high-dimensional settings. The methods are shown to be more effective in exploiting the local geometry of the parameter space, leading to faster convergence and better mixing of the Markov chain. The paper also highlights the importance of proper metric selection and the benefits of using position-specific metrics in MCMC sampling.The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo (MALA and HMC) methods defined on Riemann manifolds to address the limitations of traditional MCMC algorithms when sampling from high-dimensional, strongly correlated target densities. These methods automatically adapt to the local structure of the parameter space, improving efficiency and convergence. The Riemann manifold framework allows for more efficient proposal mechanisms by incorporating the geometry of the target density, reducing the need for manual tuning of proposal densities. The methods are evaluated on logistic regression, log-Gaussian Cox point processes, stochastic volatility models, and Bayesian dynamic systems. The results show significant improvements in time-normalized effective sample size compared to alternative methods. MATLAB code is provided for replication. Key concepts include the Fisher-Rao metric tensor, geometric concepts in MCMC, and the use of Riemannian geometry to define the metric tensor. The paper demonstrates how Riemann manifold MALA (MMALA) and Riemann manifold HMC (RMHMC) methods can be implemented to improve sampling efficiency in high-dimensional settings. The methods are shown to be more effective in exploiting the local geometry of the parameter space, leading to faster convergence and better mixing of the Markov chain. The paper also highlights the importance of proper metric selection and the benefits of using position-specific metrics in MCMC sampling.