We investigate a new methodology, worked out by Ledoux and Massart,
to prove concentration-of-measure inequalities. The method
is based on certain modified logarithmic Sobolev inequalities.
We provide some very simple and general ready-to-use inequalities.
One of these inequalities may be considered as an exponential
version of the Efron--Stein inequality.
The main purpose of this paper is to point out
the simplicity and the generality of the approach.
We show how the new method can recover many of
Talagrand's revolutionary inequalities and provide
new applications in a variety of problems including
Rademacher averages, Rademacher chaos, the number of
certain small subgraphs in a random graph, and
the minimum of the empirical risk in some statistical
estimation problems.