Fra\"iss\'e limits of C*-algebras

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.