Particle filters are a powerful and flexible tool for performing inference on state-space models. They involve a collection of samples evolving over time through a combination of sampling and re-sampling steps. The re-sampling step is necessary to ensure that weight degeneracy is avoided. In several situations of statistical interest, it is important to be able to compare the estimates produced by two different particle filters; consequently, being able to efficiently couple two particle filter trajectories is often of paramount importance. In this text, we propose several ways to do so. In particular, we leverage ideas from the optimal transportation literature. In general, though computing the optimal transport map is extremely computationally expensive, to deal with this, we introduce computationally tractable approximations to optimal transport couplings. We demonstrate that our resulting algorithms for coupling two particle filter trajectories often perform orders of magnitude more efficiently than more standard approaches.