Full waveform inversion has many virtues and a couple of glaring defects. The computational expense defect is slowly disappearing in the face of algorithm, software, and hardware advances. Another more mathematically fundamental defect is the tendency of iterative FWI algorithms to stagnate at uninformative models ("cycle skipping"). Suggestions for cycle-skipping "cures" began almost as soon as FWI was fully formulated in the 1980's and creative and promising ideas continue to appear. I will review some of these suggestions, emphasizing commonalities and differences of algorithmic structure, as well as aspects of the relation between earth model parameters and data prediction that appear to explain some of the features of these algorithms.