How do the roots of a monic polynomial whose coefficients depend smoothly on parameters depend on those parameters? This question plays an important role in many areas, e.g., PDEs, perturbation theory of linear operators, and local smooth analysis. The subject, which started with Rellich’s work in the 1930s, enjoyed sustained interest through time that intensified in the last two decades, bringing some definitive optimal results. In particular, if the coefficients are of class $C^{d-1,1}$, where $d$ is the degree of the polynomial, then any continuous root is of Sobolev class $W^{1,q}$, for each $1 \le q < d/(d-1)$. In this talk, I will discuss this optimal result and recent work investigating the continuity of the solution map “coefficients-to-roots”. Joint work with Adam Parusinski