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Deep learning has become a breathtaking technology in the last years, overcoming traditional handcrafted approaches and even humans for many different tasks. However, in some tasks, such as the verification of handwritten signatures, the amount of publicly available data is scarce, what makes difficult to test the real limits of deep learning. In addition to the lack of public data, it is not easy to evaluate the improvements of novel proposed approaches as different databases and experimental protocols are usually considered. The main contributions of this study are: i) we provide an in-depth analysis of state-of-the-art deep learning approaches for on-line signature verification, ii) we present and describe the new DeepSignDB on-line handwritten signature biometric public database, iii) we propose a standard experimental protocol and benchmark to be used for the research community in order to perform a fair comparison of novel approaches with the state of the art, and iv) we adapt and evaluate our recent deep learning approach named Time-Aligned Recurrent Neural Networks (TA-RNNs) for the task of on-line handwritten signature verification. This approach combines the potential of Dynamic Time Warping and Recurrent Neural Networks to train more robust systems against forgeries. Our proposed TA-RNN system outperforms the state of the art, achieving results even below 2.0% EER when considering skilled forgery impostors and just one training signature per user.