We will present a Kantorovich-dual Theorem that involves new opimal transport costs. This general notion of transport cost encompasses many costs used in the litterature, including the classical one introduced by Talagrand and Marton in the 90'. As a by-product, we have a full description of a large class of transport-entropy inequalities in terms of exponential integrability of some new infimum convolution operators. Applications and explicit examples of discrete measures satisfying these new weak transport-entropy inequalities will be given. Based on joint work with Nathael Gozlan, Cyril Roberto and Prasad Tetali.