They were invented by Ronald Jensen for his proof that cardinal transfer theorems hold under the axiom of constructibility.
Whilst it is possible to define so-called gap-n morasses for n > 1, they are so complex that focus is usually restricted to the gap-1 case, except for specific applications.
[1][2] Velleman[2] and Shelah and Stanley[3] independently developed forcing axioms equivalent to the existence of morasses, to facilitate their use by non-experts.
Other variants on morasses, generally with added structure, have also appeared over the years.
Velleman's clear definition can be found in,[9] where he also constructed (ω0,1) simplified morasses in ZFC.