  We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite
II$_{1}$ factor as Fra\"iss\'e limits of suitable classes of structures.
Moreover by means of Fra\"iss\'e theory we provide new examples of AF algebras
with strong homogeneity properties. As a consequence of our analysis we deduce
Ramsey-theoretic results about the class of full-matrix algebras.
