In classical peak interpolation the setting is a subalgebra A of C(K), the continuous scalar functions on a compact Hausdorff space K, and one tries to build functions in A which have prescribed values or behaviour on a fixed closed subset E of K (or on several disjoint subsets). The sets E that `work' for this are the p-sets, namely the closed sets whose characteristic functions are in the `second annihilator' (or weak* closure) of A. Glicksberg's peak set theorem characterizes these sets as the intersections of peak sets, i.e. sets F for which there is a norm 1 function f in A which is 1 exactly on F. The typical peak interpolation result, originating in results of E. Bishop, says that if f is a strictly positive function in C(K), then the continuous functions on E which are restrictions of functions in A, and which are dominated by the `control function' f on E, have extensions h in A with |h(x)| dominated by f(x) on all of K. It also yields `Urysohn type lemmas' in which we find functions in A which are 1 on E and close to zero on a closed set disjoint from E. We discuss our generalizations of these results where A above is replaced by an algebra of operators on a Hilbert space, and open and closed sets are replaced by Akemann's noncommutative topology. In fact we have recently been able to essentially complete this theory. This is mostly joint work, with Damon Hay, Matt Neal, and Charles Read.