We present DeepTract, a deep-learning framework for estimation of white matter fibers orientation and streamline tractography. We take a data-driven approach for fiber reconstruction from raw diffusion MRI, without assuming a specific diffusion model. We use a recurrent neural network for mapping sequences of diffusion-weighted imaging (DWI) values into probabilistic fiber orientation distributions. Based on these estimations, our model can perform both deterministic and probabilistic tractography on unseen DWI datasets. We quantitatively evaluate our method using the Tractometer tool, demonstrating comparable performance to state-of-the-art classical and DL-based methods. We further present qualitative results of bundle-specific probabilistic tractography of our method.