We obtain some optimal inequalities on tail probabilities for sums of independent random variables. For bounded summands, the results complete an upper bound on tail probabilities due to Talagrand. For non bounded summands, we improve Bennett's inequality under Bernstein's condition in the spirit of Hoeffding and Talagrand. In particular, we obtain a one-term asymptotic expansion for large deviations, which can be regarded as sharp large deviation result of types of Cramér and Bahadur-Rao. This asymptotic expansion improves the classical standard of Bernstein.