Joint work with K. Ball, F. Barthe and A. Naor. I will show that if X_1, X_2, ... are independent and identically distributed square-integrable random variables then the entropy of the normalized sum Ent (X_1+ ... + X_n)/\sqrt{n} is an increasing function of n. This resolves an old problem which goes back to Shannon (1949) and was formally stated as a conjecture by Lieb in 1978. The result also has a version for non-identically distributed random variables or random vectors.