We consider spread-out models of self-avoiding walk, bond percolation,
lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long
finite-range connections, above their upper critical dimensions $d=4$
(self-avoiding walk), $d=6$ (percolation) and $d=8$ (trees and animals).
The two-point functions for 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 0 to x, and the generating function
for lattice trees or lattice animals containing 0 and x. We use the
lace expansion to prove that for sufficiently spread-out models above
the upper critical dimension, the two-point function of each model
decays, at the critical point, as a multiple of $|x|^{2-d}$ as $x \to
\infty$. We use a new unified method to prove convergence of the lace
expansion. The method is based on x-space methods rather than the
Fourier transform. Our results also yield unified and simplified proofs
of the bubble condition for self-avoiding walk, the triangle condition
for percolation, and the square condition for lattice trees and lattice
animals, for sufficiently spread-out models above the upper critical
dimension.