Given two oriented, closed, connected manifolds, we can consider the set of mapping degrees of continuous maps between these manifolds. This raises the following question: Which subsets of Z can arise in this way? We present a construction by Neofytidis-Wang-Wang and Costoya-Munoz-Viruel that all finite subsets (containing 0) are mapping degree sets. Moreover, we will construct an explicit example of a subset that is not.