  In this paper we look at Grothendieck's work on classifying holomorphic
bundles over the complex projective line. The paper is divided into $4$ parts.
The first and second part we build up the necessary background to talk about
vector bundles, sheaves, cohomology, etc. The main result of the $3^{rd}$
chapter is the classification of holomorphic vector bundles over the complex
projective line. In the $4^{th}$ chapter we introduce principal $G$-bundles and
some of the theory behind them and finish off by proving Grothendieck's theorem
in full generality. The goal is a (mostly) self-contained proof of
Grothendieck's result accessible to someone who has taken differential
geometry.
