In this article, the authors aim to maximally utilize multimodality neuroimaging and genetic data for identifying Alzheimer's disease (AD) and its prodromal status, Mild Cognitive Impairment (MCI), from normal aging subjects. Multimodality neuroimaging data such as MRI and PET provide valuable insights into brain abnormalities, while genetic data such as single nucleotide polymorphism (SNP) provide information about a patient's AD risk factors. When these data are used together, the accuracy of AD diagnosis may be improved. However, these data are heterogeneous (e.g., with different data distributions), and have different number of samples (e.g., with far less number of PET samples than the number of MRI or SNPs). Thus, learning an effective model using these data is challenging. To this end, we present a novel three-stage deep feature learning and fusion framework, where deep neural network is trained stage-wise. Each stage of the network learns feature representations for different combinations of modalities, via effective training using the maximum number of available samples. Specifically, in the first stage, we learn latent representations (i.e., high-level features) for each modality independently, so that the heterogeneity among modalities can be partially addressed, and high-level features from different modalities can be combined in the next stage. In the second stage, we learn joint latent features for each pair of modality combination by using the high-level features learned from the first stage. In the third stage, we learn the diagnostic labels by fusing the learned joint latent features from the second stage. To further increase the number of samples during training, we also use data at multiple scanning time points for each training subject in the dataset. We evaluate the proposed framework using Alzheimer's disease neuroimaging initiative (ADNI) dataset for AD diagnosis, and the experimental results show that the proposed framework outperforms other state-of-the-art methods.