Non-extremal intersecting p-brane solutions of gravity coupled with se
veral antisymmetric fields and dilatons in various space-time dimensio
ns are constructed. The construction uses the same algebraic method of
finding solutions as in the extremal case and modified ''no-force'' c
onditions. We justify the ''deformation'' prescription. It is shown th
at the nonextremal intersecting p-brane solutions satisfy the harmonic
superposition rule and intersections of non-extremal p-branes are spe
cified by the same characteristic equations for the incidence matrices
as for the extremal p-branes. We show that S-duality holds for non-ex
tremal p-brane solutions. Generalized T-duality takes place under addi
tional restrictions to the parameters of the theory, which are the sam
e as in the extremal case. (C) 1997 Published by Elsevier Science B.V.