The recent trend in vision-based multi-object tracking (MOT) is heading towards leveraging the representationalpower of deep learning to jointly learn to detect and trackobjects. However, existing methods train only certain sub-modules using loss functions that often do not correlate withestablished tracking evaluation measures such as Multi-Object Tracking Accuracy (MOTA) and Precision (MOTP). As these measures are not differentiable, the choice of appropriate loss functions for end-to-end training of multi-object tracking methods is still an open research problem. In this paper, we bridge this gap by proposing a differentiable proxy of MOTA and MOTP, which we combine in a loss function suitable for end-to-end training of deep multi-object trackers. As a key ingredient, we propose a DeepHungarian Net (DHN) module that approximates the Hungarian matching algorithm. DHN allows estimating the correspondence between object tracks and ground truth objects to compute differentiable proxies of MOTA and MOTP, which are in turn used to optimize deep trackers directly. We experimentally demonstrate that the proposed differentiable framework improves the performance of existing multi-object trackers, and we establish a new state of the art on the MOTChallenge benchmark. Our code is publicly available from https://github.com/yihongXU/deepMOT.