New transportation cost inequalities are derived by means of elementary large deviation reasonings. Their dual characterization is proved; this provides an extension of a well-known result of S. Bobkov and F. Götze. Their tensorization properties are investigated. Sufficient conditions (and necessary conditions too) for these inequalities are stated in terms of the integrability of the reference measure. Applying these results leads to new deviation results: concentration of measure and deviations of empirical processes.