  We study Daugavet points and $\Delta$-points in Lipschitz-free Banach spaces.
We prove that, if $M$ is a compact metric space, then $\mu\in S_{\mathcal
F(M)}$ is a Daugavet point if, and only if, there is no denting point of
$B_{\mathcal F(M)}$ at distance strictly smaller than two from $\mu$. Moreover,
we prove that if $x$ and $y$ are connectable by rectifiable curves of lenght as
close to $d(x,y)$ as we wish, then the molecule $m_{x,y}$ is a $\Delta$-point.
Some conditions on $M$ which guarantee that the previous implication reverses
are also obtained. As a consequence of our work, we show that Lipschitz-free
spaces are natural examples of Banach spaces where we can guarantee the
existence of $\Delta$-points which are not Daugavet points.
