In the existing literature, characteristic sets in constructive commutative algebra are defined in two slightly different ways. The definitions of J.F. Ritt and W.T. Wu do not fully coincide, and this ambiguity causes a lot of confusion. In this report, the difference between these two approaches is emphasized. It is shown that a Ritt-characteristic set fully determines a prime polynomial ideal. In the irreducible case, the structure of characteristic sets and the link between the definitions of Ritt and Wu is completely determined. This leads in the reducible case to a method for the decomposition of an arbitrary variety into irreducible parts, and to an algorithm to solve the membership problem for radical ideals.