We investigate the linear-response conductance through a pair of coupled quantum dots. The conductance spectrum under ideal conditions is shown to consist of two sets of twin peaks whose locations and amplitudes are determined by the interdot coupling and the intradot charging. We will show that the qualitative features of the spectrum survive against experimental nonidealities such as (1) detuning of the individual dots, (2) interdot charging, (3) inelastic scattering, and (4) multiple lateral states. The effect of higher lateral states depends strongly on the nature of the interaction potential, screening lengths, and exchange terms, but the lowest set of twin peaks is largely unaffected by these details.