The classical Kakeya needle problem asks: what is the smallest set in the plane inside which a unit-length needle can be translated and rotated through a full 360-degree turn? In this talk, we will show how to construct such sets of arbitrarily small measures (known as Kakeya sets). We will then describe some applications to several important problems in analysis where this seemingly innocuous rotation property can lead to very counterintuitive and pathological results.