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.