  We show that among all Musielak-Orlicz function spaces on a $\sigma$-finite
non-atomic complete measure space equipped with either the Luxemburg norm or
the Orlicz norm the only spaces with the Daugavet property are $L_1$,
$L_{\infty}$, $L_1\oplus_1 L_{\infty}$ and $L_1\oplus_{\infty} L_{\infty}$. We
obtain in particular complete characterizations of the Daugavet property in the
weighted interpolation spaces, the variable exponent Lebesgue spaces (Nakano
spaces) and the Orlicz spaces.
