The present work reports some new aspects of non-axisymmetric gravity currents obtained from laboratory experiments, fully resolved simulations and box models. Following the earlier work [Zgheib et al. 2014 Theor. Comput. Fluid Dyn. 28, 521-529] which demonstrated that gravity currents initiating from non-axisymmetric cross-sectional geometries do not become axisymmetric, nor do they retain their initial shape during the slumping and inertial phases of spreading, we show that such non-axisymmetric currents eventually reach a self-similar regime during which (i) the local front propagation scales as t^(1/2) as in circular releases and (ii) the non-axisymmetric front has a self-similar shape that primarily depends on the aspect ratio of the initial release. Complementary experiments of non-Boussinesq currents and top-spreading currents suggest that this self-similar dynamics is independent of the density ratio, vertical aspect ratio, wall friction, and Reynolds number, provided Re is large, i.e., Re≥Ο(10^4). The local instantaneous front Froude number obtained from the fully-resolved simulations is compared to existing models of Froude functions. The recently reported extended box model (EBM) is capable of capturing the dynamics of such non-axisymmetric flows. Here we use the EBM to propose a relation for the self-similar horizontal aspect ratio χ_∞ of the propagating front as a function of the initial horizontal aspect ratioχ_0, namely χ_∞=1+(1/3)ln χ_0. The experimental and numerical results are in good agreement with the proposed relation.