  There is only limited experimental evidence for the existence in nature of
phase transitions of Ehrenfest order greater than two. However, there is no
physical reason for their non-existence, and such transitions certainly exist
in a number of theoretical models in statistical physics and lattice field
theory. Here, higher-order transitions are analysed through the medium of
partition function zeros. Results concerning the distributions of zeros are
derived as are scaling relations between some of the critical exponents.
