The Daugavet property in the Musielak-Orlicz spaces

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 $L_1$, $L_{\mathrm{\infty }}$, $L_1{\oplus }_1{L}_{\mathrm{\infty }}$ and $L_1{\oplus }_{\mathrm{\infty }}{L}_{\mathrm{\infty }}$. 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.