The paper proposes Metropolis-adjusted Langevin and Hamiltonian Monte Carlo (MCMC) sampling methods defined on Riemann manifolds to address the limitations of existing MCMC algorithms when sampling from high-dimensional and strongly correlated target densities. These methods provide fully automated adaptation mechanisms, eliminating the need for costly pilot runs to tune proposal densities. The Riemannian geometry of the parameter space is exploited to adaptively adjust to local structures, leading to more efficient sampling even in high dimensions. The performance of these Riemannian manifold MCMC methods is evaluated through inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models, and Bayesian estimation of dynamic systems described by nonlinear differential equations. The methods show substantial improvements in time-normalized effective sample size compared to alternative sampling approaches. MATLAB code is available for replication.The paper proposes Metropolis-adjusted Langevin and Hamiltonian Monte Carlo (MCMC) sampling methods defined on Riemann manifolds to address the limitations of existing MCMC algorithms when sampling from high-dimensional and strongly correlated target densities. These methods provide fully automated adaptation mechanisms, eliminating the need for costly pilot runs to tune proposal densities. The Riemannian geometry of the parameter space is exploited to adaptively adjust to local structures, leading to more efficient sampling even in high dimensions. The performance of these Riemannian manifold MCMC methods is evaluated through inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models, and Bayesian estimation of dynamic systems described by nonlinear differential equations. The methods show substantial improvements in time-normalized effective sample size compared to alternative sampling approaches. MATLAB code is available for replication.