The Universe is not homogenous. Since the early times, its structures have grown and moved under the laws of gravity. By measuring these motions today we are able to trace the spatial distribution of dark matter and accurately map the Universe. The comparison of the recovered large scale structures with the predictions of the concordance model of cosmology is then a powerful test of the laws of expansion and gravity.
Peculiar velocity measurements however rely on noisy extragalactic distances estimation, which can be provided by Type Ia Supernovae observations. Because peculiar velocities of galaxies are deviations from the Hubble expansion law, they have poor signal to noise ratio and are particularly sensible to systematic uncertainties such as calibration errors, selection effects and unknown covariances. During this presentation I will present how to measure peculiar velocities and how to map the Universe with them. I will detail how the statistical methods I develop can handle extragalactic distance measurements and lead to a velocity field reconstruction of our Local Universe so as constraints on cosmology. I will then show how future SNeIa surveys, in particular the Zwicky Transient Facility and the Large Synoptic Sky Survey, will allow us to accurately test General Relativity.