Let $K$ and $L$ be two convex bodies in ${\bb R}^n$ . The volume ratio ${\rm vr}(K, L)$ of $K$ and $L$ is defined by ${\rm vr}(K, L) = \inf(\vert K\vert/\vert T(L)\vert)^{1/n}$ , where the infimum is over all affine transformations $T$ of ${\bb R}^n$ for which $T(L) \subseteq K$ . It is shown in this paper that ${\rm vr}(K, L) \leqslant c \sqrt{n} \log n$ , where $c > 0$ is an absolute constant. This is optimal up to the logarithmic term.