In the past two years, over 30 papers have proposed to use convolutional neural network (CNN) for AD classification. However, the classification performances across studies are difficult to compare. Moreover, these studies are hardly reproducible because their frameworks are not publicly accessible. Lastly, some of these papers may reported biased performances due to inadequate or unclear validation procedure and also it is unclear how the model architecture and parameters were chosen. In the present work, we aim to address these limitations through three main contributions. First, we performed a systematic literature review of studies using CNN for AD classification from anatomical MRI. We identified four main types of approaches: 2D slice-level, 3D patch-level, ROI-based and 3D subject-level CNN. Moreover, we found that more than half of the surveyed papers may have suffered from data leakage and thus reported biased performances. Our second contribution is an open-source framework for classification of AD. Thirdly, we used this framework to rigorously compare different CNN architectures, which are representative of the existing literature, and to study the influence of key components on classification performances. On the validation set, the ROI-based (hippocampus) CNN achieved highest balanced accuracy (0.86 for AD vs CN and 0.80 for sMCI vs pMCI) compared to other approaches. Transfer learning with autoencoder pre-training did not improve the average accuracy but reduced the variance. Training using longitudinal data resulted in similar or higher performance, depending on the approach, compared to training with only baseline data. Sophisticated image preprocessing did not improve the results. Lastly, CNN performed similarly to standard SVM for task AD vs CN but outperformed SVM for task sMCI vs pMCI, demonstrating the potential of deep learning for challenging diagnostic tasks.