Spatio-spectral fusion of panchromatic (PAN) and hyperspectral (HS) images is of great importance in improving spatial resolution of images acquired by many commercial HS sensors. DenseNets have recently achieved great success for image super-resolution because they facilitate gradient flow by concatenating all the feature outputs in a feedforward manner. In this article, we propose a residual hyper-dense network (RHDN) that extends the DenseNet to solve the spatio-spectral fusion problem. The overall structure of the proposed RHDN method is a two-branch network, which allows the network to capture the features of HS images within and outside the visible range separately. At each branch of the network, a two-stream strategy of feature extraction is designed to process PAN and HS images individually. A convolutional neural network (CNN) with cascade residual hyper-dense blocks (RHDBs), which allows direct connections between the pairs of layers within the same stream and those across different streams, is proposed to learn more complex combinations between the HS and PAN images. The residual learning is adopted to make the network efficient. Extensive benchmark evaluations well demonstrate that the proposed RHDN fusion method yields significant improvements over many widely accepted state-of-the-art approaches.