A longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then $K_X+mL$ is basepoint free for $m>=n+1$. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for $n>=2$ the conjecture holds for m larger than $n(loglog(n)+3)$. This is joint work with L. Ghidelli.