This paper extends the multilevel Monte Carlo variance reduction technique to
nonlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particle
filter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transport
methods to re-establish correlation between coarse and fine ensembles after resampling; this controls
the variance of the estimator. Numerical examples present a proof of concept of the effectiveness
of the proposed method, demonstrating significant computational cost reductions (relative to the
single-level ETPF counterpart) in the propagation of ensembles.