New CO interferometer data show that the molecular gas in infrared ultraluminous galaxies is in rotating nuclear disks or rings. The CO maps yield disk radii, kinematic major axes, rotation speeds, enclosed dynamical masses, and gas masses. The CO brightness temperatures, the double-peaked CO line profiles, the limits on thermal continuum flux from dust, and the constraint that the gas mass must be less than the dynamical mass all indicate that the CO lines are subthermally excited and moderately opaque (τ = 4 to 10). We fit kinematic models in which most of the CO flux comes from a moderate-density warm intercloud medium, rather than from self-gravitating clouds. Typical ring radii are 300 to 800 pc. We derive gas masses not from a standard CO-to-mass ratio, but from a model of radiative transfer through subthermally excited CO in the molecular disks. This model yields gas masses of ~5 × 109 M☉, ~5 times lower than the standard method, and a ratio M/L ≈ 0.8 M☉ (K km s-1 pc2)-1. In the nuclear disks, we derive a ratio of gas to dynamical mass of Mgas/Mdyn ≈ 1/6, and a maximum ratio of gas to total mass surface density, μ/μtot, of 1/3. For the galaxies VII Zw 31, Arp 193, and IRAS 10565+2448, the CO position-velocity diagrams provide good evidence for rotating molecular rings with a central gap. In addition to the rotating central rings or disks, a new class of star formation region is identified, which we call an extreme starburst. These have a characteristic sizes of only 100 pc, with about 109 M☉ of gas and an IR luminosity of ≈ 3 × 1011 L☉ from recently formed OB stars. Four extreme starbursts are identified in the 3 closest galaxies in the sample, including Arp 220, Arp 193, and Mrk 273. These are the most prodigious star formation events in the local universe, each representing about 1000 times as many OB stars as 30 Doradus. In Mrk 231, the CO (2-1) velocity diagram along the line of nodes shows a 12 diameter inner disk and a 3'' diameter outer disk. The narrow CO line width, the single-peak line profile, the equality of the major and minor axes, and the observed velocity gradients all imply that the molecular disk is nearly face-on, yielding low optical and UV extinction to the active galactic nucleus (AGN). Such a geometry means that the molecular disk cannot be heated by the AGN; the far-infrared (FIR) luminosity of Mrk 231 is powered by a starburst, not the AGN. In Mrk 273, the CO (1-0) maps show long streamers of radius 5 kpc (7'') with velocity gradients north-south, and a nuclear disk of radius 400 pc (06) with velocity gradients east-west. The nuclear disk contains a bright CO core of radius 120 pc (02). In Arp 220, the CO and 1.3 mm continuum maps show the two nuclei embedded in a central ring or disk at P.A. 50° and a fainter structure extending 7'' (3 kpc) to the east, normal to the nuclear disk. Models of the CO and dust flux indicate that the two K-band sources contain high-density gas, with n(H2) = 2 × 104 cm-3. There is no evidence that these sources really are the premerger nuclei. They are more likely to be compact extreme starburst regions, containing 109 M☉ of dense molecular gas and new stars, but no old stars. Most of the HCN emission arises in the two nuclei. The luminosity-to-mass ratios for the CO sources in Arp 220 are compatible with the early phases of compact starbursts. There is a large mass of molecular gas currently forming stars with plenty of ionizing photons, and no obvious AGN. The entire bolometric luminosity of Arp 220 comes from starbursts, not an AGN. The CO maps show that the gas in ultraluminous IR galaxies is in extended disks that cannot intercept all the power of central AGNs, even if they exist. We conclude that in ultraluminous IR galaxies—even in Mrk 231, which hosts a quasar—the FIR luminosity is powered by extreme starbursts in the molecular rings or disks, not by dust-enshrouded quasars.