The tempered (generic) L-packet conjecture predicts that every tempered L-packet of a quasisplit group over a local field contains a generic member (as stated in my 1990 Annals paper). This conjecture has played a crucial role in the Langlands program thus far. On the other hand, Arthur's endoscopic classification of representations of classical groups determines non-tempered Arthur packets, both globally and locally, for these groups. This suggests extensions of the tempered (generic) L-packets conjecture to the non-tempered setting. In this talk, I will discuss these extensions and provide some evidence. The conjectures are naturally in terms of "Wave Fronts", the singularities of characters of the representations in the packets, an important and active area of research. I conclude by mentioning some global consequences. The conjectures include one formulated jointly with A. Hazeltine, B. Liu, and C.-H. Lo, together with one conjectured by D. Jiang.