After recalling the main properties of a conformal embedding of Lie algebras g⊃p, which is defined by the equality of the Sugawara central charges on both sides, we launch a systematic study of their branching rules. The bulk of the paper is devoted to the proof of a general formula in the case su(mn)1⊃su(m)n⊕su(n)m. At the end we give some applications to the construction of modular invariant partition functions. © 1990 Springer-Verlag.