There have been incredible advances in our ability to represent perturbative scattering amplitudes in terms of gauge-invariant, physical information. An important insight leading to this progress has been the realization that the still-difficult problem of loop integration can and should be separated from the (now understood to be) much simpler problem of constructing loop integrands. I review and update the powerful unitarity-based methods to construct integrands constructively, and argue that these new representations should make loop integration easier.
In the second part of the talk, I discuss evidence testing this conviction. In particular, I will show that many loop integrals turn out to be much easier than many had feared; but that these easy (polylogarithmic) integrals, unfortunately, represent a set of measure zero. I will conclude with a survey of recent results addressing the question of what kinds of functions are needed to represent general amplitudes perturbative quantum field theory?