Important advances have been made using convolutional neural network (CNN) approaches to solve complicated problems in areas that rely on grid structured data such as image processing and object classification. Recently, research on graph convolutional neural networks (GCNN) has increased dramatically as researchers try to replicate the success of CNN for graph structured data. Unfortunately, traditional CNN methods are not readily transferable to GCNN, given the irregularity and geometric complexity of graphs. The emerging field of GCNN is further complicated by research papers that differ greatly in their scope, detail, and level of academic sophistication needed by the reader. The present paper provides a review of some basic properties of GCNN. As a guide to the interested reader, recent examples of GCNN research are then grouped according to techniques that attempt to uncover the underlying topology of the graph model and those that seek to generalize traditional CNN methods on graph data to improve prediction of class membership. Discrete Signal Processing on Graphs (DSPg) is used as a theoretical framework to better understand some of the performance gains and limitations of these recent GCNN approaches. A brief discussion of Topology Adaptive Graph Convolutional Networks (TAGCN) is presented as an approach motivated by DSPg and future research directions using this approach are briefly discussed.