The Falconer Conjecture says that if $E$ is a compact set in $\mathbb{R}^d$ with Hausdorff dimension larger than $d/2$, then its distance set, consisting of all distinct distances generated by points in $E$, should have strictly positive Lebesgue measure. This conjecture remains open in all dimensions $d \geq 2$. In this talk we will discuss several recent developments on it, which are based on joint works with Xiumin Du, Larry Guth, Alex Iosevich, Hong Wang, Bobby Wilson, and Ruixiang Zhang.