Hyperspectral images are well-known for their fine spectral resolution to discriminate different materials. However, their spatial resolution is relatively low due to the trade-off in imaging sensor technologies, resulting in limitations in their applications. Inspired by recent achievements in convolutional neural network (CNN) based super-resolution (SR) for natural images, a novel three-dimensional full CNN (3D-FCNN) is constructed for spatial SR of hyperspectral images in this paper. Specifically, 3D convolution is used to exploit both the spatial context of neighboring pixels and spectral correlation of neighboring bands, such that spectral distortion when directly applying traditional CNN based SR algorithms to hyperspectral images in band-wise manners is alleviated. Furthermore, a sensor-specific mode is designed for the proposed 3D-FCNN such that none of the samples from the target scene are required for training. Fine-tuning by a small number of training samples from the target scene can further improve the performance of such a sensor-specific method. Extensive experimental results on four benchmark datasets from two well-known hyperspectral sensors, namely hyperspectral digital imagery collection experiment (HYDICE) and reflective optics system imaging spectrometer (ROSIS) sensors, demonstrate that our proposed 3D-FCNN outperforms several existing SR methods by ensuring higher quality both in reconstruction and spectral fidelity.