We study how the geometry of a projective variety X is reflected in the positivity properties of the diagonal ΔX considered as a cycle on X×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 Hk,0(X) vanish for k>0. We also classify varieties of low dimension where the diagonal is nef and big.