Hyperspectral image (HSI) sharpening, which aims at fusing an observable low spatial resolution (LR) HSI (LR-HSI) with a high spatial resolution (HR) multispectral image (HR-MSI) of the same scene to acquire an HR-HSI, has recently attracted much attention. Most of the recent HSI sharpening approaches are based on image priors modeling, which are usually sensitive to the parameters selection and time-consuming. This paper presents a deep HSI sharpening method (named DHSIS) for the fusion of an LR-HSI with an HR-MSI, which directly learns the image priors via deep convolutional neural network-based residual learning. The DHSIS method incorporates the learned deep priors into the LR-HSI and HR-MSI fusion framework. Specifically, we first initialize the HR-HSI from the fusion framework via solving a Sylvester equation. Then, we map the initialized HR-HSI to the reference HR-HSI via deep residual learning to learn the image priors. Finally, the learned image priors are returned to the fusion framework to reconstruct the final HR-HSI. Experimental results demonstrate the superiority of the DHSIS approach over existing state-of-the-art HSI sharpening approaches in terms of reconstruction accuracy and running time.