  In 1924, S. Banach and A. Tarski proved an astonishing, yet rather
counterintuitive paradox: given a solid ball in $\mathbb{R}^3$, it is possible
to partition it into finitely many pieces and reassemble them to form two solid
balls, each identical in size to the first. When this paradox is applied to
3-dimensional space it does go against our intuition, but very often our
intuition is flawed.
  The aim of the paper is to provide a comprehensive proof of the Banach-Tarski
paradox, expanding in between the lines of the original volume. We explore the
notions of paradoxical and equidecomposable sets which are phrased in terms of
group actions. Finally, provided we have the Axiom of Choice at our disposal,
we can construct sets that are nonmeasurable (not Lebesgue measurable) and the
proof of the Banach-Tarski Paradox follows naturally.
