A set is said to be amorphous if it is infinite, but is not the disjoint union of two infinite subsets. Thus amorphous sets can exist only if the axiom of choice is false. We give a general study of the structure which an amorphous set can carry, with the object of eventually obtaining a complete classification. The principal types of amorphous set we distinguish are the following: amorphous sets not of projective type, either bounded or unbounded size of members (...) of partitions of the set into finite pieces), and amorphous sets of projective type, meaning that the set admits a non-degenerate pregeometry, over finite fields either of bounded cardinality or of unbounded cardinality. The hope is that all amorphous sets will be of one of these types. Examples of each sort are constructed, and a reconstruction result for bounded amorphous sets is presented, indicating that the amorphous sets of this kind constructed in the paper are the only possible ones. The final section examines some questions concerned with the resulting cardinal arithmetic. (shrink)