  We study the Daugavet property in tensor products of Banach spaces. We show
that $L_1(\mu)\widehat{\otimes}_\varepsilon L_1(\nu)$ has the Daugavet property
when $\mu$ and $\nu$ are purely non-atomic measures. Also, we show that
$X\widehat{\otimes}_\pi Y$ has the Daugavet property provided $X$ and $Y$ are
$L_1$-preduals with the Daugavet property, in particular spaces of continuous
functions with this property. With the same tecniques, we also obtain
consequences about roughness in projective tensor products as well as the
Daugavet property of projective symmetric tensor products.
