  We derive several tools for classifying tensor ideals in monoidal categories.
We use these results to classify tensor ideals in Deligne's universal
categories RepO, RepGL and RepP. These results are then used to obtain new
insight into the second fundamental theorem of invariant theory for the
algebraic supergroups of types A,B,C,D,P.
  We also find short proofs for the classification of tensor ideals in RepS and
in the category of tilting modules for SL2(k) with char(k)>0 and for Uq(sl2)
with q a root of unity. In general, for a simple Lie algebra g of type ADE, we
show that the lattice of such tensor ideals for Uq(g) corresponds to the
lattice of submodules in a parabolic Verma module for the corresponding affine
Kac-Moody algebra.
