  A Banach space $X$ is said to have the Daugavet property if every operator
$T: X\to X$ of rank~$1$ satisfies $\|Id+T\| = 1+\|T\|$. We show that then every
weakly compact operator satisfies this equation as well and that $X$ contains a
copy of $\ell_{1}$. However, $X$ need not contain a copy of $L_{1}$. We also
study pairs of spaces $X\subset Y$ and operators $T: X\to Y$ satisfying
$\|J+T\|=1+\|T\|$, where $J: X\to Y$ is the natural embedding. This leads to
the result that a Banach space with the Daugavet property does not embed into a
space with an unconditional basis. In another direction, we investigate spaces
where the set of operators with $\|Id+T\|=1+\|T\|$ is as small as possible and
give characterisations in terms of a smoothness condition.
