Conjunctive grammars are basically context-free grammars with an explicit set intersection operation added to the formalism of rules. This paper presents a cubic-time recognition and parsing algorithm for this family of grammars, which is applicable to an arbitrary conjunctive grammar without any initial transformations. The algorithm is in fact an extension of the context-free recognition and parsing algorithm due to Graham, Harrison and Ruzzo, and it retains the cubic time complexity of its prototype. It is shown that for the case of linear conjunctive grammars this algorithm can be modified to work in quadratic time and use linear space. The given algorithm is then applied to solve the membership problem for conjunctive grammars in polynomial time, and subsequently to prove the problem's P-completeness, as well as P-completeness of the membership problem for linear conjunctive grammars.