Random polynomials are innocent-looking objects of study in Mathematics that remain mysterious despite decades of intensive research. We will discuss the history and recent progress in the study of random polynomials. We will focus on the universality property and the distribution of the number of real roots: its mean, variance, and limiting distribution. We then discuss several intriguing open problems.

This talk is based on several joint papers with Mei-Chu Chang, Yen Do, Hoi Nguyen, and Van Vu.