In this paper, properties of a homogeneous Bose gas with a Feshbach resonance are studied in the dilute region at zero temperature. The stationary state contains condensations of atoms and molecules. The ratio of the molecule density to the atom density is $\ensuremath{\pi}{\mathrm{na}}^{3}.$ There are two types of excitations, molecular excitations and atomic excitations. Atomic excitations are gapless, consistent with the traditional theory of a dilute Bose gas. The molecular excitation energy is finite in the long-wavelength limit as observed in recent experiments on
85Rb. In addition, the decay process of the condensate is studied. The coefficient of the three-body recombination rate is about 140 times larger than that of a Bose gas without a Feshbach resonance, in reasonably good agreement with the experiment on
23Na.We model collisionless collective conversion of a degenerate Fermi gas of atoms into bosonic molecules via a Feshbach resonance, treating the bosonic molecules as a classical field and seeding the pairing amplitudes with random phases. A dynamical instability of the Fermi sea against association with molecules drives the conversion. The model qualitatively reproduces several experimental observations [Regal et al., Nature (London), (2003)]. We predict that the initial temperature of the Fermi gas sets the limit for the efficiency of atom-molecule conversion.