In this paper we derive conditions for geometric ergodicity of the random-walk-based Metropolis algorithm on . We show that at least exponentially light tails of the target density is a necessity. This extends the one-dimensional result of Mengersen and Tweedie (1996, Ann. Statist. 24, 101–121). For super-exponential target densities we characterize the geometrically ergodic algorithms and we derive a practical sufficient condition which is stable under addition and multiplication. This condition is especially satisfied for the class of densities considered in Roberts and Tweedie (1996, Biometrika 83, 95–110).