We determine the weight spectra of the Reed-Muller codes $RM(m-3,m)$ for $m\ge 6$ and $RM(m-4,m)$ for $m\ge 8$. The technique used is induction on $m$, using that the sum of two weights in $RM(r-1,m-1)$ is a weight in $RM(r,m)$, and using the characterization by Kasami and Tokura of the weights in $RM(r,m)$ that lie between its minimum distance $2^{m-r}$ and the double of this minimum distance. We also derive the weights of $RM(3,8),\,RM(4,9),$ by the same technique. We conclude with a conjecture on the weights of $RM(m-c,m)$, where $c$ is fixed and $m$ is large enough.