The extremal process of branching Brownian motion (BBM)--- i.e., the collection of particles furthest from the origin-- has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point-- the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.