We consider nearest-neighbor self-avoiding walk, bond percolation, lattice
trees, and bond lattice animals on ${\mathbb{Z}}^d$. The two-point functions of
these models are respectively the generating function for self-avoiding walks
from the origin to $x\in{\mathbb{Z}}^d$, the probability of a connection from
the origin to $x$, and the generating functions for lattice trees or lattice
animals containing the origin and $x$. Using the lace expansion, we prove that
the two-point function at the critical point is asymptotic to
$\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$, for $d\geq 5$ for self-avoiding
walk, for $d\geq19$ for percolation, and for sufficiently large $d$ for lattice
trees and animals. These results are complementary to those of [Ann. Probab. 31
(2003) 349--408], where spread-out models were considered. In the course of the
proof, we also provide a sufficient (and rather sharp if $d>4$) condition under
which the two-point function of a random walk on ${{\mathbb{Z}}^d}$ is
asymptotic to $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$.