The Earth's climate over the past million years has been characterized by a succession of cold and warm periods, known as glacial-interglacial cycles, with periodicities corresponding to those of the Earth's main orbital parameters; precession (23 kyr), obliquity (41 kyr) and eccentricity (100 kyr). The astronomical theory of climate, in which the orbital variations are taken to drive the climate changes, has been very successful in explaining many features of the palaeoclimate records. Nevertheless, the timing of the main glacial and interglacial periods remains puzzling in many respects. In particular, the main glacial-interglacial switches occur approximately every 100 kyr, but the changes in insolation forcing are very small in this frequency band. Similarly, an especially warm interglacial episode, about 400,000 years ago, occurred at a time when insolation variations were minimal. Here I propose that multiple equilibria in the climate system can provide a resolution of these problems within the framework of astronomical theory. I present two simple models that successfully simulate each glacial-interglacial cycle over the late Pleistocene epoch at the correct time and with approximately the correct amplitude. Moreover, in a simulation over the past 2 million years, the onset of the observed prominent ~100-kyr cycles around 0.8 to 1 million years ago is correctly reproduced.