This paper presents a precise and detailed study of the complexities of the membership and the universal membership problems for Abstract Categorial Grammars (ACG). ACGs have been introduced by Philippe de Groote in  as a simplification over categorial grammars which reduces the number of necessary primitives involved in the definition of the formalism. Thus in ACGs, every structure is represented with the help of the linear
-calculus and the languages defined by means of ACGs are sets of linear
-terms. The problem under investigation has already been studied in , but we give here some more precise results with some arguably simpler proofs. We use the same classification of the grammars in terms of the order of the lexicon and of the order of abstract language.