Let $k$ be a field and $X$ a geometrically connected variety over $k$. The Tate or degeneracy locus of a $l$-adic local system on $X$ is the etale counterpart of the Hodge locus of a VHS. While in the last decade tremendous progresses have been made in understanding the latter thanks to, in particular, techniques from o-minimality, much less is known about the former. I will review the main conjectures (and mention briefly some applications) about this locus when k is a number field, and explain what we can currently prove. If time remains, I will sketch some of the proofs. This should include joint works with Jakob Stix and Akio Tamagawa.