  This paper develops a theory for completely random measures in the framework
of free probability. A general existence result for free completely random
measures is established, and in analogy to the classical work of Kingman it is
proved that such random measures can be decomposed into the sum of a purely
atomic part and a (freely) infinitely divisible part. The latter part (termed a
free L\'evy basis) is studied in detail in terms of the free L\'evy-Khintchine
representation and a theory parallel to the classical work of Rajput and
Rosinski is developed. Finally a L\'evy-It\^o type decomposition for general
free L\'evy bases is established.
