By means of the numerical renormalization group method, I study the Kondo effect and quantum phase transitions in a double quantum dot system with on-site repulsion U , interdot repulsion V , and magnetic field B. At zero temperature and B = 0, there is a local spin triplet-singlet transition of the Kosterlitz-Thouless type when V increases to a critical value V c ≈ U . With increasing B, a singlet-triplet transition of the second order occurs at a critical magnetic field B c ≈ V − U . The magnetic susceptibility and the first derivatives of the spin correlation and double occupation are expected to diverge at B c and to have different critical exponents, which are universal for different parameters. An appropriate magnetic field can restore the Kondo effect.