Deep learning has shown remarkable results in every field, especially in the biomedical field, due to its ability to exploit large-scale datasets. A convolutional neural network (CNN) is a widely used deep learning approach to solve medical imaging problems. Over the past few years, many studies have focused on CNN-based techniques for brain tumor diagnosis. There are, however, still some critical challenges that CNNs face towards clinic application. This study presents a comprehensive review of current literature that involves CNN architectures for brain tumor classification. We compare the key achievements in the performance evaluation metrics of the applied classification algorithms. In addition, this review assesses the clinical effectiveness of the included studies to elaborate on the limitations and directions of this area for future work. No review focusing on the clinical effectiveness of previous works in this field has been published. We believe that this study has the potential to elevate the application of CNN-based deep learning methods in clinical practice and also can be a quick reference for biomedical researchers who are interested in this field.