Brain tumor is one of the most dangerous cancers in people of all ages, and its grade recognition is a challenging problem for radiologists in health monitoring and automated diagnosis. Recently, numerous methods based on deep learning have been presented in the literature for brain tumor classification (BTC) in order to assist radiologists for a better diagnostic analysis. In this overview, we present an in-depth review of the surveys published so far and recent deep learning-based methods for BTC. Our survey covers the main steps of deep learning-based BTC methods, including preprocessing, features extraction, and classification, along with their achievements and limitations. We also investigate the state-of-the-art convolutional neural network models for BTC by performing extensive experiments using transfer learning with and without data augmentation. Furthermore, this overview describes available benchmark data sets used for the evaluation of BTC. Finally, this survey does not only look into the past literature on the topic but also steps on it to delve into the future of this area and enumerates some research directions that should be followed in the future, especially for personalized and smart healthcare.