Single-modality medical images generally do not contain enough information to reach an accurate and reliable diagnosis. For this reason, physicians commonly rely on multimodal medical images for comprehensive diagnostic assessments. This study introduces a deep evidential fusion framework designed for segmenting multimodal medical images, leveraging the Dempster-Shafer theory of evidence in conjunction with deep neural networks. In this framework, features are first extracted from each imaging modality using a deep neural network, and features are mapped to Dempster-Shafer mass functions that describe the evidence of each modality at each voxel. The mass functions are then corrected by the contextual discounting operation, using learned coefficients quantifying the reliability of each source of information relative to each class. The discounted evidence from each modality is then combined using Dempster's rule of combination. Experiments were carried out on a PET-CT dataset for lymphoma segmentation and a multi-MRI dataset for brain tumor segmentation. The results demonstrate the ability of the proposed fusion scheme to quantify segmentation uncertainty and improve segmentation accuracy. Moreover, the learned reliability coefficients provide some insight into the contribution of each modality to the segmentation process.