How cells maintain nuclear shape and position against various intracellular and extracellular forces is not well understood, although defects in nuclear mechanical homeostasis are associated with a variety of human diseases. We estimated the force required to displace and deform the nucleus in adherent living cells with a technique to locally pull the nuclear surface. A minimum pulling force of a few nanonewtons--far greater than typical intracellular motor forces--was required to significantly displace and deform the nucleus. Upon force removal, the original shape and position were restored quickly within a few seconds. This stiff, elastic response required the presence of vimentin, lamin A/C, and SUN (Sad1p, UNC-84)-domain protein linkages, but not F-actin or microtubules. Although F-actin and microtubules are known to exert mechanical forces on the nuclear surface through molecular motor activity, we conclude that the intermediate filament networks maintain nuclear mechanical homeostasis against localized forces.