In this paper, we present a deep learning method, DDMReg, for fast and accurate registration between diffusion MRI (dMRI) datasets. In dMRI registration, the goal is to spatially align brain anatomical structures while ensuring that local fiber orientations remain consistent with the underlying white matter fiber tract anatomy. To the best of our knowledge, DDMReg is the first deep-learning-based dMRI registration method. DDMReg is a fully unsupervised method for deformable registration between pairs of dMRI datasets. We propose a novel registration architecture that leverages not only whole brain information but also tract-specific fiber orientation information. We perform comparisons with four state-of-the-art registration methods. We evaluate the registration performance by assessing the ability to align anatomically corresponding brain structures and ensure fiber spatial agreement between different subjects after registration. Experimental results show that DDMReg obtains significantly improved registration performance. In addition, DDMReg leverages deep learning techniques and provides a fast and efficient tool for dMRI registration.