Measurements of the Hubble constant, and more generally measurements of the expansion rate and distances over the interval $0 < z < 1$, appear to be inconsistent with the predictions of the standard cosmological model ($\Lambda$CDM) given observations of cosmic microwave background temperature and polarization anisotropies. Here we consider a variety of types of departures from $\Lambda$CDM that could, in principle, restore concordance among these datasets, and we explain why we find almost all of them unlikely to be successful. We single out the set of solutions that increase the expansion rate in the decade of scale factor expansion just prior to recombination as the least unlikely. These solutions are themselves tightly constrained by their impact on photon diffusion and on the gravitational driving of acoustic oscillations of the modes that begin oscillating during this epoch -- modes that project on to angular scales that are very well measured. We point out that a general feature of such solutions is a residual to fits to $\Lambda$CDM, like the one observed in Planck power spectra. This residual drives the modestly significant inferences of angular-scale dependence to the matter density and anomalously high lensing power, puzzling aspects of a data set that is otherwise extremely well fit by $\Lambda$CDM.