Consider a quantum field theory in $d$ dimensions with a symmetry $G$. For the purpose of this discussion let's say that $d$ is even and $G$ is a compact, connected Lie group. We say that the symmetry has a 't Hooft anomaly if it is not possible to gauge $G$. To detect this we couple the theory to a background gauge field $A$ for the symmetry $G$ and look for gauge dependence of the partition function. Variation of $\mathrm{log} \ Z(A)$ with gauge parameter $\epsilon$ defines a functional $I(\epsilon, A)$. It is a simple exercise to check that this functional has to satisfy the so called Wess-Zumino consistency conditions.
Solutions of the consistency condition can be constructed using the method of descent: we start from an invariant polynomial in curvature in $(d+2)$-dimensions, to which one can associate a Chern-Simons theory in $(d+1)$-dimensions. Then the gauge variation of the Chern-Simons action on a $(d+1)$-dimensional manifold with a boundary is a local functional on the boundary (which is of dimension $d$) and satisfies the consistency condition for the same reason as the sought after $I(\epsilon, A)$ does.
It is my impression that many people believe that these solutions of the consistency condition are the only ones (or perhaps at least the only ones which are relevant in the context of anomalies). Is this correct? If so, why?