A Kakeya Set in $\mathbb{R}^n$ is a compact set $E\subset\mathbb{R}^n$ containing a unit line segment in every direction, possessing, in other words, the Kakeya Property: $$\forall\, e\in S^{n-1} \; \exists\, x\in E\; \forall\, t\in [-1/2,1/2]\hspace{.3 cm}x+te \in E$$ The Kakeya Problem — the question of what measure constraints the Kakeya Property imposes — remains largely open. I'll discuss its solution in some restricted contexts, particularly when $n=2$, presenting several constructions of "small" planar Kakeya sets and the maximality of such sets' Hausdorff dimension. I'll note the difficulty of extending those approaches to the general case, and the centrality of the problem to core questions in harmonic analysis.