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.