We explore the computability of Julia sets under the computable analysis model. We begin by showing that Julia sets without Siegel discs are always computable. As for Julia sets with Siegel discs, Braverman and Yampolsky used a result of Buff and Cheritat in complex dynamics to show that Julia sets with Siegel discs are sometimes noncomputable. I will give an overview of their results, introducing concepts from complex dynamics and computable analysis as we go along.