The classical network p - median problem is the problem of identifying p (supply) points of the network minimizing some suitable measure of the collective "distances" of these p points from a finite number of designated possible "demand" points. This p - median problem has been generalized in this dissertation to the case where demand is continuously distributed, and then discussed in terms of points that are natural and distinguishing in the network context, i.e., "proper intersection" points and "interior" points, thus avoiding the less useful "vertices" and "nodes" terminologies commonly found in the literature;For the generalized 1 - median problem with uniformly distributed demand, and with the objective of minimizing distance, it is shown that membership of a "circuit" by an "edge" is sufficient for the disqualification of the interior points of the edge as possible supply point locations. For other versions, where demand is distributed in other than uniform fashion, and/or the objective is either a minimization of a suitable function of distance, or a minimization of travel cost, conditions are given to disqualify interior points of edges;For the generalized p - median problem (p (GREATERTHEQ) 2), with uniformly distributed demand and with the objective of minimizing distance, it is shown that the interior points of p edges belonging to p "disjoint minimal circuits" can be disqualified;As an example, a particular highway maintenance garage location problem has been modeled and analyzed as a generalized p - median problem.