In the usual notation, the Bombieri-Vinogradov theorem states that $\sum_{q\leq Q}\max_{(a,q)=1}\max_{y\le x}| \psi (y,q,a)-y/\varphi (q)| \ll x(\log x)^{-A} \tag{*}$ holds with $$Q=\sqrt{x}(\log x)^{-B}$$, for any fixed $$A\ge 0$$ with a suitable constant $$B=B(A)$$. Improving on recent results of R. C. Vaughan and B. V. Levin, the author shows that one may take $$B=A+11/8$$ in (*). His result constitutes the currently sharpest form of (*). Attention should be drawn here to recent work of E. Bombieri, J. B. Friedlander and H. Iwaniec (see the preceding review) who showed that $$Q$$ can be taken substantially larger than $$\sqrt{x}$$ in (*), provided the maximum over $$(a,q)=1$$ is dropped in the sum and only residue classes a mod q with a fixed integer a are considered.