Statistics of roots of congruences

The roots of quadratic congruences are a classical object of study, and largely due to applications in analytic number theory, their statistical distribution is well-studied. On the other hand, very little is known about roots of higher degree congruences beyond their equidistribution. We discuss some recent results (joint with Jens Marklof and Zonglin Li) in the quadratic case and some results related to the cubic case.