In this paper we review and discuss some fundamental aspects of the random version of the Olami-Feder- Christensen model, and its relevance for the understanding of self-organized criticality (SOC). We review the universal character of the exponent <img width=32 height=32 id="_x0000_i1026" src="../../img/revistas/bjp/v33n3/a09img01.gif" align=absmiddle > or = 3/2, related to avalanche size distributions in random SOC models, and its connection to branching processes theory. We also generalize previous results, that had been obtained for the random OFC model with four neighbors, to any coordination number. Finally we present some connections between our generalization and recent discussions involving the branching rate approach to this model.