This paper addresses the problem of atomic multicasting messages in asynchronous distributed systems. Firstly, we give a characterization of the notion of "genuine atomic multicast''. This characterization leads to a better understanding of the difference between atomic multicast and atomic broadcast, and to make a clear distinction between genuine atomic multicast algorithms and non-genuine atomic multicast algorithms. Secondly, we consider a system with at least two processes among which one can crash, and we show that, in contrast to atomic broadcast, genuine atomic multicast is impossible to solve with failure detectors that are unreliable, i.e., that cannot distinguish crashed processes from correct ones. Finally, we discuss a way to circumvent the impossibility result, by restricting the destinations of multicasts to sets of disjoint process groups, each group behaving like a logically correct entity.