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

Completely Random Measures and L\'evy Bases in Free probability