  We characterise narrow and strong Daugavet operators on $C(K,E)$-spaces;
these are in a way the largest sensible classes of operators for which the norm
equation $\|Id+T\| = 1+\|T\|$ is valid. For certain separable range spaces $E$
including all finite-dimensional ones and locally uniformly convex ones we show
that an unconditionally pointwise convergent sum of narrow operators on
$C(K,E)$ is narrow, which implies for instance the known result that these
spaces do not have unconditional FDDs. In a different vein, we construct two
narrow operators on $C([0,1],\ell_1)$ whose sum is not narrow.
