The gist of the theory is that the fiber functor Φ of the Galois theory is replaced by an exact and faithful tensor functor F from C to the category of finite-dimensional vector spaces over K. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group G of natural transformations of F into itself, that respect the tensor structure.
Let K be a field and C a K-linear abelian rigid tensor (i.e., a symmetric monoidal) category such that
Then C is a Tannakian category (over K) if there is an extension field L of K such that there exists a K-linear exact and faithful tensor functor (i.e., a strong monoidal functor) F from C to the category of finite dimensional L-vector spaces.
A Tannakian category over K is neutral if such exact faithful tensor functor F exists with L=K.
Morally[clarification needed], the philosophy of motives tells us that the Hodge structure and the Galois representation associated to an algebraic variety are related to each other.
Mumford-Tate conjecture proposes that the algebraic groups arising from the Hodge strucuture and the Galois representation by means of Tannakian categories are isomorphic to one another up to connected components.
Another place in which Tannakian categories have been used is in connection with the Grothendieck–Katz p-curvature conjecture; in other words, in bounding monodromy groups.