  In this paper we prove two new cases of Langlands functoriality. The first is
a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms
on $GL_6$, from which we obtain our second case, the long awaited functorial
symmetric cube map for cusp forms on $GL_2$. We prove these by applying a
recent version of converse theorems of Cogdell and Piatetski-Shapiro to
analytic properties of certain $L$-functions obtained from the method of
Eisenstein series (Langlands-Shahidi method). As a consequence, we prove the
bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at
every place, finite or infinite, breaking the crucial bound 1/6 (see below and
Section 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for
$GL_2$. Many other applications are obtained.
