One thing people (myself included!) often missed though is that Metropolis is extremely prone to flicker. I think the best summary of why comes from Kalos and Whitlock (2nd Edition):
> The M(RT)2 algorithm is very simple and powerful; it can be used to sample essentially any distribution function regardless of analytic complexity in any number of dimensions. Complementary disadvantages are that sampling is correct only asymptotically and that successive variables produced are correlated, often very strongly. This means that the evaluation of integrals normally produces positive correlations in the values of the integrand, with consequent increase in variance for a fixed number of steps as compared with independent samples. Also the method is not well suited to sampling distributions with parameters that change frequently.
> The M(RT)2 algorithm is very simple and powerful; it can be used to sample essentially any distribution function regardless of analytic complexity in any number of dimensions. Complementary disadvantages are that sampling is correct only asymptotically and that successive variables produced are correlated, often very strongly. This means that the evaluation of integrals normally produces positive correlations in the values of the integrand, with consequent increase in variance for a fixed number of steps as compared with independent samples. Also the method is not well suited to sampling distributions with parameters that change frequently.