Diffusion tensor imaging (DTI) is widely used to examine the human brain white matter structures, including their microarchitecture integrity and spatial fiber tract trajectories. It has clinical applications in several neurological disorders and neurosurgical guidance. However, a major factor that prevents DTI from being incorporated in clinical routines is its long scan time due to the acquisition of a large number (typically 30 or more) of diffusion-weighted images (DWIs) required for reliable tensor estimation. Here, a deep learning-based technique has been developed to obtain tensor-derived quantitative maps and fiber tractography with only six DWIs, resulting in a significant reduction in imaging time. The method uses deep convolutional neural networks to learn the nonlinear relationship between DWIs and several tensor-derived maps, bypassing the conventional tensor fitting procedure, which is well known to be highly susceptible to noises in DWIs. The performance of the method was evaluated using DWI datasets from the Human Connectome Project and patients with ischemic stroke. Our results demonstrate that the proposed technique is able to generate fractional anisotropy (FA) and mean diffusivity (MD) maps, as well as fiber tractography, from as few as six DWIs. With results from 90 DWIs as the ground truth, the proposed method from six DWIs achieves a quantification error of less than 3% in all regions of interest in white matter structures and 15% in gray matter structures. In addition, we also demonstrate that the neural network trained using healthy volunteers can be directly applied/tested on stroke patients' DWIs data without compromising the lesion detectability. Such a significant reduction in scan time will allow the inclusion of DTI into clinical routine for many potential applications.