An adaptive Metropolis (AM) algorithm is introduced, which updates the Gaussian proposal distribution based on the full history of the process. Unlike traditional Metropolis-Hastings algorithms, the AM algorithm is non-Markovian but maintains correct ergodic properties. The algorithm adapts continuously to the target distribution, adjusting both the size and spatial orientation of the proposal distribution. It is easy to implement and has been shown to perform well in numerical tests, often matching or outperforming traditional methods. The AM algorithm uses the covariance matrix of the sampled states to update the proposal distribution, ensuring that it approaches a scaled Gaussian approximation of the target distribution. This improves the efficiency of the simulation. The algorithm's ergodicity is proven under the assumption that the target density is bounded and has bounded support. The AM algorithm is tested on various target distributions, including Gaussian and nonlinear ones, and is shown to perform well in simulations. The results indicate that the AM algorithm is a reliable and effective method for Markov chain Monte Carlo simulations.An adaptive Metropolis (AM) algorithm is introduced, which updates the Gaussian proposal distribution based on the full history of the process. Unlike traditional Metropolis-Hastings algorithms, the AM algorithm is non-Markovian but maintains correct ergodic properties. The algorithm adapts continuously to the target distribution, adjusting both the size and spatial orientation of the proposal distribution. It is easy to implement and has been shown to perform well in numerical tests, often matching or outperforming traditional methods. The AM algorithm uses the covariance matrix of the sampled states to update the proposal distribution, ensuring that it approaches a scaled Gaussian approximation of the target distribution. This improves the efficiency of the simulation. The algorithm's ergodicity is proven under the assumption that the target density is bounded and has bounded support. The AM algorithm is tested on various target distributions, including Gaussian and nonlinear ones, and is shown to perform well in simulations. The results indicate that the AM algorithm is a reliable and effective method for Markov chain Monte Carlo simulations.