  We study how the geometry of a projective variety $X$ is reflected in the
positivity properties of the diagonal $\Delta_X$ considered as a cycle on $X
\times X$. We analyze when the diagonal is big, when it is nef, and when it is
rigid. In each case, we give several implications for the geometric properties
of $X$. For example, when the diagonal is big, we prove that the Hodge groups
$H^{k,0}(X)$ vanish for $k>0$. We also classify varieties of low dimension
where the diagonal is nef and big.
