  We consider directed first-passage and last-passage percolation on the
nonnegative lattice Z_+^d, d\geq2, with i.i.d. weights at the vertices. Under
certain moment conditions on the common distribution of the weights, the limits
g(x)=lim_{n\to\infty}n^{-1}T(\lfloor nx\rfloor) exist and are constant a.s. for
x\in R_+^d, where T(z) is the passage time from the origin to the vertex z\in
Z_+^d. We show that this shape function g is continuous on R_+^d, in particular
at the boundaries. In two dimensions, we give more precise asymptotics for the
behavior of g near the boundaries; these asymptotics depend on the common
weight distribution only through its mean and variance. In addition we discuss
growth models which are naturally associated to the percolation processes,
giving a shape theorem and illustrating various possible types of behavior with
output from simulations.
