This thesis represents a culmination of work and learning that has taken place over a period of almost three years (2007 - 2010) at the University of Salerno, and at the University of Connecticut. It is mostly an unified mathematical dissertation of the running consensus procedures. In the recent years, the detection using the paradigm of the running consensus has been recognized as one of the three possible classes of distributed detection in which the phases of sensing and communication need not be mutually exclusive, i.e., sensing and communication occur simultaneously. Considering that the running consensus paradigm is just an intuitive inference procedure, i.e. sub-optimal w.r.t. an ideal centralized system scheme which is optimal, the most important result is that it asymptotically reaches the performance of this ideal scheme.