  The lace expansion has been a powerful tool for investigating mean-field
behavior for various stochastic-geometrical models, such as self-avoiding walk
and percolation, above their respective upper-critical dimension. In this
paper, we prove the lace expansion for the Ising model that is valid for any
spin-spin coupling. For the ferromagnetic case, we also prove that the
expansion coefficients obey certain diagrammatic bounds that are similar to the
diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk.
As a result, we obtain Gaussian asymptotics of the critical two-point function
for the nearest-neighbor model with d>>4 and for the spread-out model with d>4
and L>>1, without assuming reflection positivity.
