Developing a breast cancer screening method is very important to facilitate early breast cancer detection and treatment. Building a screening method using medical imaging modality that does not cause body tissue damage (non-invasive) and does not involve physical touch is challenging. Thermography, a non-invasive and non-contact cancer screening method, can detect tumors at an early stage even under precancerous conditions by observing temperature distribution in both breasts. The thermograms obtained on thermography can be interpreted using deep learning models such as convolutional neural networks (CNNs). CNNs can automatically classify breast thermograms into categories such as normal and abnormal. Despite their demostrated utility, CNNs have not been widely used in breast thermogram classification. In this study, we aimed to summarize the current work and progress in breast cancer detection based on thermography and CNNs. We first discuss of breast thermography potential in early breast cancer detection, providing an overview of the availability of breast thermal datasets together with publicly accessible. We also discuss characteristics of breast thermograms and the differences between healthy and cancerous thermographic patterns. Breast thermogram classification using a CNN model is described step by step including a simulation example illustrating feature learning. We cover most research related to the implementation of deep neural networks for breast thermogram classification and propose future research directions for developing representative datasets, feeding the segmented image, assigning a good kernel, and building a lightweight CNN model to improve CNN performance. INDEX TERMS breast cancer; convolutional neural network; deep learning; early detection; thermogram