We consider extensions of Hoeffding's " exponential method " approach for obtaining upper estimates on the probability that a sum of independent and bounded random variables is significantly larger than its mean. We show that the exponential function in Hoeffding's approach can be replaced with any function which is non-negative, increasing and convex. As a result we generalize and improve upon Hoeffding's inequality. Our approach allows to obtain " missing factors " in Hoeffding's inequality. The later result is a rather weaker version of a theorem that is due to Michel Talagrand. Moreover, we characterize the class of functions with respect to which our method yields optimal concentration bounds. Finally, using ideas from the theory of Bernstein polynomials, we show that similar ideas apply under information on higher moments of the random variables.