This thesis explores the Kakeya conjecture for n=2, which states that every subset of Rⁿ containing a unit line segment in every direction has Minkowski dimension n. To tackle this problem we explore what the Minkowski dimension is, and use the Kakeya maximal operator. We start by stating the idea of a Kakeya needle set. We construct a sequence of these sets with arbitrarily small measure, followed by looking at Kakeya sets, which can even have measure equal to zero. We derive a proof of the Kakeya conjecture using the Kakeya maximal operator conjecture and its dual form, where we also prove all necessary implications. Resulting in a complete proof for n=2 with respect to the Minkowski dimension.