In the 1920s Besicovitch asked the question: What can one say about the structure of sets $E$ in the plane, with the property that $\lim_{r \downarrow 0} \frac{ \mathcal{H}^{1}(B(x,r) \cap E)}{2r}= 1$ for almost every $x \in E$? In 1987 Preiss gave a complete answer to Besicovitch's density question in groundbreaking work relying on his introduction of the notion of tangent measures. In the first half of this talk, I will introduce tangent measures and some of their properties that make them widely applicable. In the second half I will briefly introduce several previous results relying on techniques from Preiss' paper and then discuss how, in work-in-progress with Tatiana Toro and Bobby Wilson, we recover anisotropic analogs of several of these results.