In the last two decades the development of Total Order (TO) broadcast and multicast communication over asynchronous distributed systems have been one of the main research issues in dependable distributed computing. As a result, a huge amount of works has been carried out, ranging from service speci.cations to a variety of TO implementations over different communication platforms. Differences among such speci.cations can make very difficult the choice of the right TO primitive, by an application designer, to enable the application to meet its correctness requirements. The aim of this paper is thus to present a clear classification of total order broadcast specifications. In particular, six specifications of total order broadcast primitives proposed in the literature are organized into a hierarchy that allows (i) to classify existing implementations of total order communication primitives, and (ii) to select the right primitive according to the application requirements in order to maximize performance.