  We analyze non-linear interactions of 2N-form Maxwell fields in a space-time
of dimension D=4N. Based on the Pasti-Sorokin-Tonin (PST) method, we derive the
general consistency condition for the dynamics to respect both manifest
SO(2)-duality invariance and manifest Lorentz invariance. For a generic
dimension D=4N, we determine a canonical class of exact solutions of this
condition, which represent a generalization of the known non-linear duality
invariant Maxwell theories in D=4. The resulting theories are shown to be
equivalent to a corresponding class of canonical theories formulated a la
Gaillard-Zumino-Gibbons-Rasheed (GZGR), where duality is a symmetry only of the
equations of motion. In dimension D=8, via a complete solution of the PST
consistency condition, we derive new non-canonical manifestly duality invariant
quartic interactions. Correspondingly, we construct new non-trivial quartic
interactions also in the GZGR approach, and establish their equivalence with
the former. In the presence of charged dyonic p-brane sources, we reveal a
basic physical inequivalence of the two approaches. The power of our method
resides in its universal character, reducing the construction of non-linear
duality invariant Maxwell theories to a purely algebraic problem.
