This talk gives a guided tour through examples and indicative theory that highlight two important topics that are typically omitted from applied mathematics or engineering training. The first is an appreciation for the calculus of distributions, that is different to the calculus of functions. That implies that inference in the presence of uncertainty is fundamentally different to inversion, and that the notion of 'best fit' can be misleading, or just plain wrong. The second is the toolbox of mid-level and high-level representations that can do a much better job than the all-too-common low-level models such as pixel representations. One has to give up many seductive comforts, such as linear spaces and projections on Hilbert spaces, but the potential gain is the ability to compute answers to questions that motivated the inverse problem, with valid quantification of uncertainties.