We show how allowing non-local terms in the field equations of symmetric tensors uncovers a neat geometry that naturally generalizes the Maxwell and Einstein cases. The end results can be related to multiple traces of the generalized Riemann curvatures Ralpha 1...alpha s; beta 1...beta s introduced by de Wit and Freedman (1980), divided by suitable powers of the D'Alembertian operator square. The conventional local equations can be recovered by a partial gauge fixing involving the trace of the gauge parameters Lambdaalpha 1...alpha s-1, absent in the Fronsdal formulation. The same geometry underlies the fermionic equations, that, for all spins s+1/2, can be linked via the operator part/square to those of the spin-s bosons.