Firstly, we propose and illustrate a refinement of Deligne?RTMs principle: every infinitesimal deformation problem over a field of characteristic zero with cohomology constraints is governed by a differential graded Lie algebra together with a module. Secondly, we review recent results about the global structure of cohomology jump loci of rank one local systems. Finally, we address future directions for other types of jump loci. All this is joint work with Botong Wang.