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We investigate a fractional notion of gradient and divergence operator. Wegeneralize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractionaldiv-curl quantities, obtaining, in particular, a nonlocal version of Wente'slemma.  We demonstrate how these quantities appear naturally in nonlocal geometricequations, which can be used to obtain a theory for fractional harmonic mapsanalogous to the local theory. Firstly, regarding fractional harmonic maps intospheres, we obtain a conservation law analogous to Shatah's conservation lawand give a new regularity proof analogous to H\'elein's for harmonic maps intospheres.  Secondly, we prove regularity for solutions to critical systems with nonlocalantisymmetric potentials on the right-hand side. Since the half-harmonic mapequation into general target manifolds has this form, as a corollary, we obtaina new proof of the regularity of half-harmonic maps into general targetmanifolds following closely Rivi\`{e}re's celebrated argument in the localcase.  Lastly, the fractional div-curl quantities provide also a new, simpler, prooffor H\"older continuity of $W^{s,n/s}$-harmonic maps into spheres and we extendthis to an argument for $W^{s,n/s}$-harmonic maps into homogeneous targets.This is an analogue of Strzelecki's and Toro-Wang's proof for $n$-harmonic mapsinto spheres and homogeneous target manifolds, respectively.

Fractional div-curl quantities and applications to nonlocal geometric equations