  The paper is devoted to establishing some general exponential inequalities
for supermartingales. The inequalities improve or generalize many exponential
inequalities of Bennett, Freedman, de la Pe\~{n}a, Pinelis and van de Geer.
Moreover, our concentration inequalities also improve some known inequalities
for sums of independent random variables. Applications associated with linear
regressions, autoregressive processes and branching processes are provided. In
particular, an interesting application of {de la Pe\~{n}a's} inequality to
self-normalized deviations is also provided.
