Abstarct: Abstract Let R be a commutative ring. The relation on R given by a~b if and only if ann(a)=ann(b) is an equivalence relation. The compressed zero-divisor graph, denoted by ΓE(R), is the graph whose vertices are the equivalence classes induced by ~ other than [0] and [1], such that two distinct vertices [a] and [b] are adjacent if and only if ab=0. In this thesis, we investigate when ΓE(R), is planar. First, we characterize all finite non-local rings whose compressed zero-divisor graphs are planar. In the local case, we characterize all graphs that are realized as the planar compressed zero-divisor graphs in some cases. Finally, we find all graphs with at most five vertices that are realized as the compressed zero-divisor graphs.