Dynamic time warping (DTW) is a popular technique for sequence alignment, and is the de facto standard for dynamic signature verification. In this paper, we go a significant step further to enhance DTW with the capability of deep representation learning, and propose an end-to-end trainable Deep soft-DTW (DsDTW) model for dynamic signature verification. Specifically, we design a convolutional recurrent adaptive network (CRAN) to process dynamic signatures, and utilize it to provide robust and discriminative local representations as inputs for DTW. As DTW is not fully differentiable with regard to its inputs, we introduce its smoothed formulation, soft-DTW, and incorporate the soft-DTW distances of signature pairs into the loss function for optimization. Because soft-DTW is differentiable, the proposed DsDTW is end-to-end trainable, and achieves an elegant integration of CRAN deep learning model and traditional DTW mechanism. Our method achieves state-of-the-art performance on several public benchmarks, and has won first place in the ICDAR 2021 competition for online signature verification. Source codes of DsDTW is available at https://github.com/KAKAFEI123/DsDTW.