Recent progress on the local Langlands conjecture for $G_2$

I will describe recent work, joint with G. Savin, in which we prove a dichotomy result for generic cuspidal representations of the exceptional group G_2, over a p-adic field.  This result reduces the local Langlands conjectures for generic cuspidal represenations G_2 to the corresponding conjectures on the classical groups PGL_3 and PGSp_6.  These corresponding conjectures are proven and "almost proven", respectively.  Our methods include a study of the theta correspondence in the groups E_6 and E_7, a uniqueness result for "Shalika periods" in PGSp_6, and various incarnations of Spin L-functions.    The talk will include much background on exceptional groups and the local Langlands conjecture, accessible to a general number theoretic audience.  I aim to present a precise summary of our results, with hints about the method of proof.