I will discuss developments for algebraic and p-adic aspects of L-functions, with a view toward my ongoing joint work with G. Rosso and S. Shah on spin L-functions for GSp_6.  I will emphasize how these developments fit into a broader context, highlighting common ingredients among constructions during the past several decades, while also indicating where new technical challenges arise.  All are welcome, and expertise in spin L-functions is unnecessary to follow this lecture.