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Convolutional neural networks (CNNs) have played an important role in medical imaging—from diagnostics to research to data integration. This has allowed clinicians to plan operations, diagnose patients earlier, and study rare diseases in more detail. However, data quality, quantity, and imbalance all pose challenges for CNN training and accuracy; in addition, training costs can be high when many types of CNNs are needed in a health care system. Topology and geometry provide tools to ameliorate these challenges for CNNs when they are integrated into the CNN architecture, particularly in the data preprocessing steps or convolution layers. This paper reviews the current integration of geometric tools within CNN architectures to reduce the burden of large training datasets and offset computational costs. This paper also identifies fertile areas for future research into the integration of geometric tools with CNNs.