The definition of a pre-Tannakian category is somewhat technical, but it is in essence an axiomatization of categories with all the same internal structure as the finite dimensional representation theory of a group G.  If a pre-Tannakian category admits a "fiber functor" to vector spaces  (a concrete realization of it as a collection of vector spaces)  then it is said to be Tannakian, and Tannakian reconstruction says that in this case it is actually equivalent to representations of some group.  However not every pre-Tannakian category is Tannakian, and Deligne constructed explicit examples Rep(St), Rep(GLt), and Rep(Ot) of categories which are pre-Tannakian but not Tannakian, which interpolate the representation theories of the symmetric, general linear, and orthogonal groups.  

I have been fascinated by these Deligne categories for years, and a lot of my work has been trying to understand and generalize them.  My first project in graduate school was to apply the Deligne category machinery to prove a conjecture about representations of Weyl groups of type B. Then I defined a new integral form of Rep(St) with better reduction to positive characteristic, which motivated my definition with Etingof and Ostrik of p-adic dimensions in symmetric tensor categories.  I also gave a new interpretation of Deligne's categories using model theory.  This perspective has proved useful for understanding the internal structure of these categories - Daniil Kalinov and I used it to characterize algebra objects in Rep(St), Flake, Laugwitz, and I used it to describe the Drinfeld center of Rep(St), and it has been utilized further in the work of Kalinov, Utiralova, Etingof, Rains, and others.  

Recently, Andrew Snowden and I found a new construction of pre-Tannakian categories using oligomorphic groups, a class of groups that normally arise in model theory.  This gives another new interpretation of Rep(St),  but also shows that the world of pre-Tannakian categories is much bigger than previously thought.  We are still actively developing this theory. Since that first paper we have: extended the oligomorphic group set-up to linear groups, joined with Noah Snyder to analyze the Delannoy Category (a particularly interesting example), and extended that analysis to a more complicated circular version,  then we introduced the oligomorphic analogs of Grothendieck's Galois categories, and used them to give a universality statement about our construction.  There's been a flurry of activity in this area, and more results are on the way.