The paper surveys recent results on the Ramanujan conjecture until the year 2004. The most general version of Ramanujans conjecture, which the author states, is that the local component \(\pi_v\) (at a place \(v\) of a global field \(F\)) of every globally generic cuspidal representation \(\Pi\) of the \(F\)-adelic points of a quasisplit connected reductive algebraic group \(G\) must be a tempered representation of the group of \(F_v\)-rational points of \(G\).
Considering the Langlands classification for an irreducible admissible local component \(\pi\), by A. J. Silberger [Math. Ann. 236, 95–104 (1978; Zbl 0362.20029)], Theorem 4.1 (3) in the \(p\)-adic case, or A. Borel and N. Wallach [Continuous cohomology, discrete subgroups, and representations of reductive groups (Ann. of Math. Stud. 94, Princeton Univ. Press 1980; Zbl 0443.22010)], Theorem 4.11 in the Archimedean case, \(\pi\) is given by a Langlands quotient parametrized by a tempered representation \(\sigma\) of a suitable Levi subgroup of \(G\) and a parameter \(\nu\) which determines a suitable character twist within a normalized parabolic induction. The conjecture \(\pi= \sigma\) and \(\nu= 0\) is now reformulated by the assertion that \(\nu\) can be arbitrarily small bounded. Indeed for \(\text{GL}_n\) the parameter \(\nu\) can be formulated as a sequence of positive real numbers parametrizing unramified twists of \(\sigma\) (see S. S. Kudla [Proc. Symp. Pure Math. 55, 365–391 (1994; Zbl 0811.11072)], Theorem 2.2.2).
The author surveys the discussion on several bounds and additional reformulations for cases other than \(\text{GL}_n\). A large part of the paper surveys results concerning the Langlands functoriality because this method, for instance, makes it possible to reduce the conjecture for classical groups to that for \(\text{GL}_n\).