Abstract: We construct a plane set of Lebesgue measure zero which contains a unit line segment in every direction. These are called Besicovitch sets. Then, we construct a plane set of arbitrary small Lebesgue measure, inside which a unit segment can be reversed, that is, maneuvered to lie in its original position but rotated through 180 degrees without leaving the set. This is called a Kakeya needle set. We follow the exposition of K. J. Falconer, "The Geometry of Fractal Sets", Cambridge University Press, 2002. In a later lecture, we review how Charles Fefferman used these ideas to refute the disc conjecture in Fourier multipliers.