We present ISO Short-Wavelength Spectrometer observations of pure-rotational line emission H 2 from the disks around low-and intermediate-mass preÈmain-sequence stars as well as from young stars thought to be surrounded by debris disks. The preÈmain-sequence sources have been selected to be isolated from molecular clouds and to have circumstellar disks revealed by millimeter interferometry. We detect "" warm ÏÏ (T B 100È200 K) gas around many sources, including tentatively the debris-disk H 2 objects. The mass of this warm gas ranges from D10~4 up to 8 ] 10~3 and can constitute a M _ M _ nonnegligible fraction of the total disk mass. Complementary single-dish 12CO 3È2, 13CO 3È2, and 12CO 6È5 observations have been obtained as well. These transitions probe cooler gas at T B 20È80 K. Most objects show a double-peaked CO emission proÐle characteristic of a disk in Keplerian rotation, consistent with interferometer data on the lower J lines. The ratios of the 12CO 3È2/13CO 3È2 integrated Ñuxes indicate that 12CO 3È2 is optically thick but that 13CO 3È2 is optically thin or at most moderately thick. The 13CO 3È2 lines have been used to estimate the cold gas mass. If a conversion H 2 /CO factor of 1 ] 104 is adopted, the derived cold gas masses are factors of 10È200 lower than those deduced from 1.3 millimeter dust emission assuming a gas/dust ratio of 100, in accordance with previous studies. These Ðndings conÐrm that CO is not a good tracer of the total gas content in disks since it can be photodissociated in the outer layers and frozen onto grains in the cold dense part of disks, but that it is a robust tracer of the disk velocity Ðeld. In contrast, can shield itself from photodissociation even in H 2 low-mass "" optically thin ÏÏ debris disks and can therefore survive longer. The warm gas is typically 1%È10% of the total mass deduced from millimeter continuum emission, but it can increase up to 100% or more for the debris-disk objects. Thus, residual molecular gas may persist into the debris-disk phase. No signiÐcant evolution in the CO, or dust masses is found for stars with ages in the range of H 2 , 106È107 yr, although a decrease is found for the older debris-disk star b Pictoris. The large amount of warm gas derived from raises the question of the heating mechanism(s). Radiation from the central H 2 star as well as the general interstellar radiation Ðeld heat an extended surface layer of the disk, but existing models fail to explain the amount of warm gas quantitatively. The existence of a gap in the disk can increase the area of material inÑuenced by radiation. Prospects for future observations with ground-and space-borne observations are discussed. Subject headings : circumstellar matter È infrared : stars È planetary systems : protoplanetary disks 1 Based in part on observations with ISO, an ESA project with instruments funded by ESA member states (especially the PI countries : France, Germany, Netherlands, and the United Kingdom) and with participation of ISAS and NASA.