The tame flows are ""nice"" flows on ""nice"" spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Phi: mathbb{R}times Xrightarrow X on pfaffian set X is tame if the graph of Phi is a pfaffian subset of mathbb{R}times Xtimes X. Any compact tame set admits plenty tame flows. The author proves that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metric is tame.