  We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand
pairs. First, we develop a criterion for a direct limit representation to be a
multiplicity--free discrete direct sum of irreducible representations. Then we
look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian
symmetric spaces, where we combine our criterion with the Cartan--Helgason
Theorem to show in general that the regular representation of $G = \varinjlim
G_n$ on a certain function space $\varinjlim L^2(G_n/K_n)$ is multiplicity
free. That method is not applicable for direct limits of nonsymmetric Gelfand
pairs, so we introduce two other methods. The first, based on ``parabolic
direct limits'' and ``defining representations'', extends the method used in
the symmetric space case. The second uses some (new) branching rules from
finite dimensional representation theory. In both cases we define function
spaces $\cA(G/K)$, $\cC(G/K)$ and $L^2(G/K)$ to which our multiplicity--free
criterion applies.
