Abstract. Let M n = X 1 + · · · + X n be a sum of independent random variables such that X k ≤ 1, E X k = 0 and E X 2 k = σ 2 k for all k. Hoeffding 1963, Theorem 3, proved thatBentkus 2004 improved Hoeffding's inequalities using binomial tails as upper bounds. Letk stand for the skewness and kurtosis of X k . In this paper we prove (improved) counterparts of the Hoeffding inequality replacing σ 2 by certain functions of γ 1 , . . . , γ n respectively κ 1 , . . . , κ n . Our bounds extend to a general setting where X k are martingale differences, and they can combine the knowledge of skewness and/or kurtosis and/or variances of X k . Up to factors bounded by e 2 /2 the bounds are final. All our results are new since no inequalities incorporating skewness or kurtosis control so far are known.