Abstract The low-energy Feshbach resonances recently observed in the inter-particle interactions of trapped ultra-cold atoms involve an intermediate quasi-bound molecule with a spin arrangement that differs from the trapped atom spins. Variations of the strength of an external magnetic field then alter the difference of the initial and intermediate state energies (i.e. the ‘detuning’). The effective scattering length that describes the low-energy binary collisions, similarly varies with the near-resonant magnetic field. Since the properties of the dilute atomic Bose–Einstein condensates (BECs) are extremely sensitive to the value of the scattering length, a ‘tunable’ scattering length suggests very interesting many-body studies. In this paper, we review the theory of the binary collision Feshbach resonances, and we discuss their effects on the many-body physics of the condensate. We point out that the Feshbach resonance physics in a condensate can be considerably richer than that of an altered scattering length: the Feshbach resonant atom–molecule coupling can create a second condensate component of molecules that coexists with the atomic condensate. Far off-resonance, a stationary condensate does behave as a single condensate with effective binary collision scattering length. However, even in the off-resonant limit, the dynamical response of the condensate mixture to a sudden change in the external magnetic field carries the signature of the molecular condensate's presence: experimentally observable oscillations of the number of atoms and molecules. We also discuss the stationary states of the near-resonant condensate system. We point out that the physics of a condensate that is adiabatically tuned through resonance depends on its history, i.e. whether the condensate starts out above or below resonance. Furthermore, we show that the density dependence of the many-body ground-state energy suggests the possibility of creating a dilute condensate system with the liquid-like property of a self-determined density.