In this talk, we discuss recent work on the Hausdorff and packing dimension of pinned distance sets in the plane. Given a point $x\in\R^2$, and a set $E\subseteq \R^2$, the pinned distance set of $E$ with respect to $x$  is $\Delta_x = \{\vert x - y\vert \mid y\in E\}$ . An important open problem is understanding the Hausdorff, and packing, dimensions of $\Delta_x(E)$ for (analytic) sets $E$. In this talk, we will discuss recent progress on this problem. We will present improved lower bounds for both the Hausdorff and packing dimensions of pinned distance sets for sets of "high" dimension, i.e., $\dim_H(E) > 1$. We also discuss recent work on the pinned distance sets of sets with "low" dimension, i.e., $\dim_H(E) < 1$.  We also discuss the computability-theoretic methods used to achieve these bounds.