Langlands functoriality is a generalization of Langlands conjectures that essentially tells us that a $L$-homomorphism

${}^L G\to {}^L H$

between the L-groups of two reductive groups over $\Q$ should induce a transfer map from automorphic representations for $G$ to automorphic representations of $H$.

It is also known (starting from Jacquet and Langlands’ work for $\GL_2$) in the case of inner forms of a group: there is an equivalence between automorphic representations on the group and automorphic representations on its inner form. This was used by Harris and Taylor to prove instances of the local Langlands correspondence for $\GL_n$ by using unitary Shimura varieties associated to some twisted (unitary) forms of $\GL_n$.