BACKGROUND: Cancer has been a leading cause of death in the United States with significant health care costs. Accurate prediction of cancers at an early stage and understanding the genomic mechanisms that drive cancer development are vital to the improvement of treatment outcomes and survival rates, thus resulting in significant social and economic impacts. Attempts have been made to classify cancer types with machine learning techniques during the past two decades and deep learning approaches more recently. RESULTS: In this paper, we established four models with graph convolutional neural network (GCNN) that use unstructured gene expressions as inputs to classify different tumor and non-tumor samples into their designated 33 cancer types or as normal. Four GCNN models based on a co-expression graph, co-expression+singleton graph, protein-protein interaction (PPI) graph, and PPI+singleton graph have been designed and implemented. They were trained and tested on combined 10,340 cancer samples and 731 normal tissue samples from The Cancer Genome Atlas (TCGA) dataset. The established GCNN models achieved excellent prediction accuracies (89.9-94.7%) among 34 classes (33 cancer types and a normal group). gene-perturbation experiments were performed on four models based on co-expression graph, co-expression+singleton, PPI graph, and PPI+singleton graphs. The co-expression GCNN model was further interpreted to identify a total of 428 markers genes that drive the classification of 33 cancer types and normal. The concordance of differential expressions of these markers between the represented cancer type and others are confirmed. Successful classification of cancer types and a normal group regardless of normal tissues' origin suggested that the identified markers are cancer-specific rather than tissue-specific. CONCLUSION: Novel GCNN models have been established to predict cancer types or normal tissue based on gene expression profiles. We demonstrated the results from the TCGA dataset that these models can produce accurate classification (above 94%), using cancer-specific markers genes. The models and the source codes are publicly available and can be readily adapted to the diagnosis of cancer and other diseases by the data-driven modeling research community.