Riemannian first-passage percolation is a continuum analogue of lattice FPP. Instead of considering a random metric on the lattice $$\mathbb{Z}^2$$, we begin with a random Riemannian metric on $$\mathbb{R}^2$$. The global structures of the two models are similar - with my advisor Janek Wehr, we have proved a shape theorem for this model, which shows that balls under the Riemannian metric grow asymptotically like Euclidean balls. However, there is also a rich local structure, since Riemannian geometry provides us with notions of curvature and geodesics, curves which (locally) minimize length. Geodesics need not always globally minimize length (e.g., great circles on the sphere), and it is an interesting and important question to identify those geodesics which do so. No geometric background will be required for this talk.