A general formulation is presented for studying the motion of buoyant vortices in a homogeneous ambient fluid. It extends the well-known Hamiltonian framework for interacting homogeneous point vortices to include buoyancy effects acting on the vortices. This is then used to systematically examine the buoyant one-, two-, and three-vortex problems. In doing so we find that two buoyant vortices may evolve either as a pair in bounded circular orbits or as two independent unbounded vortices that drift apart, and a criterion is found to distinguish these cases. Special attention is given to the buoyant vortex couple, consisting of two vortices of equal and opposite circulation and equal buoyancy anomaly. We show that a theoretical maximum height is generally possible for the rise (or fall) of such couples against buoyancy forces. Finally, the possibility and onset of chaotic motions in the buoyant three-vortex problem are addressed. In contrast to the homogeneous three-vortex problem, the buoyant vortex system shows evidence that chaos is present. We also demonstrate the chaotic advection of tracer parcels arising from the flow field induced by just two buoyant vortices.