It has long been known that smooth crossovers between “Higgs” regimes and “confining” regimes are possible in some gauge theories with fundamental representation matter, leading to a notion of Higgs/confinement “complementarity”.  But the validity, or otherwise, of such complementarity can be subtle, especially in massless phases with Goldstone bosons.  This talk will examine the presence or absence of Higgs/confinement phase transitions, when there are no distinguishing local order parameters, in both instructive models and in dense QCD.