a c$^*$-algebra consists of an algebra of bounded linear operators acting on a hilbert space which is closed the adjoint operation (roughly, the transpose) and is complete in a certain metric. typical examples include the ring of $n \times n$ complex matrices and the ring $c(x)$ of representation of continuous functions from a compact space $x$ to the complex numbers. many more interesting examples arise from various dynamical objects (e.g. group and group actions) and from various geometric/topological constructions. the structure of finite dimensional c*-algebras is well understood: they are finite direct sums of complex matrix algebras. the class of approximately finite-dimensional (af) c*-algebras, ones which may be written as (the closure of) an increasing union of f1inite-dimensional subalgebras, are also well understood: they are determined up to isomorphism by their module structure. however, the class of subalgebras of af-algebras is still rather mysterious; it includes, for instance, all commutative c*-algebras and all c*-algebras generated by amenable groups. it is a long-standing problem to find an abstract characterization of subalgebras of af-algebras. i will discuss the af-embedding problem for c$^*$-algebras and a recent partial solution to this problem which gives a nearly complete characterization of c$^*$-subalgebras of simple af-algebras.