We study time correlations of last passage percolation (LPP), a model in the Kardar-Parisi-Zhang universality class, with three different geometries: step, flat and stationary. We prove the convergence of the covariances of the LPP at two different times to a limiting expression given in terms of Airy processes.
Furthermore, we prove the behaviour of the covariances when the two times are close to each other, conjectured in a work of Ferrari and Spohn.