The authors reprove and generalize the (stable) branching rules for classical symmetric pairs. In other words, given an irreducible finite dimensional representation of a classical group \(G\), they decompose its restriction to a symmetric subgroup \(H\) into irreducibles. The multiplicities of this decomposition are given in terms of the Littlewood-Richardson coefficients \(c^\lambda_{\mu\nu}\). Here \(\mu\), \(\nu\) and \(\lambda\) are integer partitions with at most \(n\) parts, and the same letters denote the corresponding representations of \(\text{GL}_n\). The coefficient \(c^\lambda_{\mu\nu}\) is the multiplicity of \(\lambda\) in \(\mu\otimes\nu\). The earliest results of this kind are due to Littlewood, for the cases \(\text{O}_n\subseteq \text{GL}_n\) and \(\text{Sp}_{2n}\subseteq \text{GL}_{2n}\). The authors of the present paper study 10 families of symmetric pairs. The main novelty in their approach is the use of the language of dual pairs developed by R. Howe. This enables them to apply a uniform approach to all 10 families under consideration. One of the main techniques in the proofs is the use of the notion of see-saw pairs of Kudla. This enables one to pass from one dual pair to another related pair.