In this thesis, we study what can be said about Fermat's Last Theorem using only elementary methods. We will study the work of Sophie Germain and Niels Henrik Abel, both who worked with Fermat’s Last Theorem in the early 1800s. Abel claimed to have proven four partial results of Fermat's Last Theorem, but mentioned nothing about how he had proved them. The aim of this thesis is to rediscover these four proofs. We will show that using only elementary methods, we are able to prove parts of Abel's theorems, but we have not found it possible to prove them in full generality. We will see that allowing ourselves to use some non-elementary methods one can prove more of Abel's claims, but still not in full generality.