The prediction of protein secondary and tertiary structures from the primary amino acid sequence is both an incredibly important and incredibly difficult problem. Accurate prediction of a protein's native structure can provide critical insights about its function, ultimately leading to breakthoughs in drug design and disease diagnosis. The field has a rich history, from the earliest folding experiments in the 1960's to the use of state-of-the-art algorithms today; this article reviews protein folding's history with an emphasis on how modern methods are tackling the protein folding problem. Assuming only a basic knowledge of biochemistry, we'll explore Christian Anfinsen's classical experiments with bovine RNase, the paradox of protein folding proposed by Cyrus Levinthal, the success of modern machine learning methods, and the promise of quantum computation for protein folding.