Branching random walk with regularly varying displacements is a heavy-tailed random field indexed by a Galton-Watson tree. We shall connect it to stable point processes introduced and characterized by Davydov, Molchanov and Zuyev (2008), who showed that such point processes can always be represented as a scale mixture of iid copies of one point process with the scaling points coming from an independent Poisson random measure. We obtain such a point process as a weak limit of a sequence of point processes induced by a branching random walk with regularly varying displacements. In particular, we show that a prediction of Brunet and Derrida (2011), remains valid in this setup, and recover a slightly improved version of a result of Durrett (1983). This talk is based on a joint work with Ayan Bhattacharya and Parthanil Roy.