First-passage percolation deals with the large-scale geometry of the randomly weighted graph obtained by placing i.i.d. non-negative weights on the edges of the standard nearest-neighbor graph on the square lattice. The “shape theorem” of Durrett and Liggett states that under mild conditions, if B(r) is the ball of radius r about the origin in the weighted graph metric, then with probability one, B(r)/r converges uniformly to a deterministic compact convex set C. In joint work with Mike Hochman and more recent work with Tuca Auffinger, we investigate the limiting shape for a broad class of distributions for which there is a coupling with an underlying oriented percolation model. We can show that in these cases the limit shape must be non-polygonal. I will explain this theorem and its various consequences, relating to infinite coexistence in stochastic growth and competition models, and, if time permits, shape fluctuation theorems.