  Riemannian first-passage percolation (FPP) is a continuum model, with a
distance function arising from a random Riemannian metric in $\R^d$. Our main
result is a shape theorem for this model, which says that large balls under
this metric converge to a deterministic shape under rescaling. As a
consequence, we show that smooth random Riemannian metrics are geodesically
complete with probability one.
