  In first-passage percolation on the integer lattice, the Shape Theorem
provides precise conditions for convergence of the set of sites reachable
within a given time from the origin, once rescaled, to a compact and convex
limiting shape. Here, we address convergence towards an asymptotic shape for
cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we
identify the asymptotic shapes associated to these graphs as restrictions of
the asymptotic shape of the lattice. Apart from providing necessary and
sufficient conditions for $L^p$- and almost sure convergence towards this
shape, we investigate also stronger notions such as complete convergence and
stability with respect to a dynamically evolving environment.
