The paper introduces an adaptive Metropolis (AM) algorithm, which updates the proposal distribution for Markov Chain Monte Carlo (MCMC) methods using all available information up to that point. Unlike traditional Metropolis-Hastings algorithms, the AM algorithm is non-Markovian but is shown to have the correct ergodic properties. The authors provide a detailed description of the algorithm, including its implementation and theoretical analysis, and demonstrate its effectiveness through numerical tests. The AM algorithm is compared with traditional Metropolis-Hastings algorithms using various target distributions, and the results indicate that the AM algorithm performs well, especially in high-dimensional cases and for nonlinear distributions. The paper also discusses the advantages of the AM algorithm over other adaptive MCMC methods and provides insights into its practical implementation.The paper introduces an adaptive Metropolis (AM) algorithm, which updates the proposal distribution for Markov Chain Monte Carlo (MCMC) methods using all available information up to that point. Unlike traditional Metropolis-Hastings algorithms, the AM algorithm is non-Markovian but is shown to have the correct ergodic properties. The authors provide a detailed description of the algorithm, including its implementation and theoretical analysis, and demonstrate its effectiveness through numerical tests. The AM algorithm is compared with traditional Metropolis-Hastings algorithms using various target distributions, and the results indicate that the AM algorithm performs well, especially in high-dimensional cases and for nonlinear distributions. The paper also discusses the advantages of the AM algorithm over other adaptive MCMC methods and provides insights into its practical implementation.