This essay discusses the challenges faced in the development of the Doctrine of Supremacy of European Union, which states that EU law takes precedence over national law. It explores the relationship between the Court of Justice and national courts concerning Supremacy, and the challenges posed by the monist and dualist approach of member states. The essay concludes that the biggest challenge faced by the Doctrine of Supremacy is the member states' consideration of the supremacy of their constitutional law over EU law.