Summary: Let \(X\) and \(W\) be two sets and \(\Delta: \overline R^X\to \overline R^W\) a duality (i.e., a mapping \(\Delta: f\in \overline R^X\to f^\Delta\in \overline R^W\) such that \((\inf_{i\in I} f_i)^\Delta= \sup_{i\in I} f^\Delta_i\) for all \(\{f_i\}_{i\in I}\subseteq \overline R^X\) and all index sets \(I\)). We introduce and study the subdifferential \(\partial^\Delta f(x_0)\) of a function \(f\in \overline R^X\) at a point \(x_0\in X\), with respect to \(\Delta\). We also consider the particular cases when \(\Delta\) is a (Fenchel-Moreau) conjugation, or a \(\vee\)-duality, or a \(\perp\)-duality.