The issues of compositionality, materialized ever since Frege (1982), are critically re-examined in language first mainly and then in all other possible representational systems such as thoughts, concept combination, computing, gesture, music, and animal cognition. The notion is regarded as necessary and suggested as neurologically correlated in humans, even if a weakened version is applicable because of non-articulated constituents and contextuality. Compositionality is crucially involved in all linguistically or non-linguistically meaningful expressions, dealing with at-issue content, default content, and even not-at-issue meanings such as implicatures and presuppositions in discourse. It is a constantly guiding principle to show the relation between representation and mind, still posing tantalizing research issues.