  Experimental evidence for the existence of strictly higher-order phase
transitions (of order three or above in the Ehrenfest sense) is tenuous at
best. However, there is no known physical reason why such transitions should
not exist in nature. Here, higher-order transitions characterized by both
discontinuities and divergences are analysed through the medium of partition
function zeros. Properties of the distributions of zeros are derived, certain
scaling relations are recovered, and new ones are presented.
