In this thesis, the ultimate aim is to present the proofs of the four classical theorems of Ramsey theory: Ramsey’s, Schur’s, van der Waerden’s, and Rado’s theorems. We discuss the finite and infinite versions of these theorems, which are equivalent to eachother, along with their proofs. Additionally, we introduce the basics of nonstandard analysis tools, so called filters. Furthermore, we present two different proofs of Schur’s and a special case of van der Waerden’s theorems using ultrafilter methods.