Spectral analysis of the cycles of glaciation shows that they consist of narrow peaks, and this strongly argues that they are astronomical in origin. However the mechanism that links astronomy to climate is uncertain. The Milankovitch theory attributes the 100 kyr cycle (which dominated for the last million years) to changes in insolation driven by changes in the earth's orbital eccentricity, and the smaller 41 kyr cycle to changes in the earth's tilt (the 'obliquity'). An alternative theory (R. Muller and G. MacDonald, Nature 1995, vol. 377, p 107-8) attributes the 100 kyr cycle to changes in orbital inclination rather than eccentricity; the linking mechanism could be accretion of meteoroids or interplanetary dust. A careful spectral analysis of diverse oxygen isotope records (Site 607, Site 806, Specmap, Site 552, Site 677) shows that the 100 kyr cycle is not compatible with the Milankovitch theory. Eccentricity should yield a triplet of peaks near 95, 125, and 400 kyr periods, but the data have only a single narrow peak near 100 kyr. Most nonlinear models (e.g. those that derive the 100 kyr cycle from the envelope of the precession curve) also predict a triplet of peaks, not present in the data. A disagreement is also seen in the bispectra. The narrowness of the observed peak is also incompatible with many free oscillation models. In contrast, the orbital inclination theory correctly predicts both the observed spectra and bispectra. For the last million years, a linear orbital inclination model accounts for about 50% of the variance of the data. Other climate proxies (e.g. the lake-bed depth ranks of Olsen and Kent, and the coarse component of Site 806) show that although eccentricity does not drive global glaciation, it can affect the local paleoclimate. For these records the triplet peak of eccentricity is clearly seen. The weaker 41 kyr glacial cycle, which dominates the oxygen isotope data in the period 1-2 Ma, may be linked to obliquity through Milankovitch insolation. It is important that the time scales of the records analyzed have not been 'tuned' to an eccentricity model, since tuning can make the spectra of any data set -- even random data -- match the Milankovitch model; this result has been independently demonstrated by B. Neeman.