I will give an overview of the theory of higher structure maps for finite free resolutions of
length 3 which saw important advances in recent years. This theory leads to some interesting
open questions, related mostly to linkage theory. I will discuss answers to some of these
questions. If time permits I will discuss similar theory for Gorenstein ideals of codimension
4. Most of these results are joint work with Lorenzo Guerrieri and Xianglong Ni.