There are some natural situations in which self-adjoint operators cannot exhibit spectral gaps for topological reasons. Perhaps the most prominent examples come from the theory of topological insulators, where boundaries very generally force the appearance of spectrum inside bulk gaps. Closely related phenomena are spectral flows, where topology can stabilize gap closings of continuous families of self-adjoint operators, leading for example to robust eigenvalue crossings. In many cases, these effects can be understood in a unified way using K-theory for C*-algebras. In this talk, I want to explain the basic mechanism behind such topological gap-filling and illustrate it through some examples.