let $m$ be a compact manifold of dimension $n \geq 3$. along the yamabe flow, a riemannian metric on $m$ is deformed according to the equation ${{\partial g}\over{\partial t}} = -(r_g - r_g) \, g$, where $r_g$ is the scalar curvature associated with the metric $g$ and $r_g$ denotes the mean value of $r_g$. it is known that the yamabe flow exists for all time. moreover, if $3 \leq n \leq 5$ or $m$ is locally conformally flat, then the solution approaches a metric of constant scalar curvature as $t \to \infty$. i will describe how this result can be generalized to dimensions $6$ and higher under a technical condition on the weyl tensor. the proof requires the construction of a suitable family of test functions.