In a generalized intersection searching problem, a set, S, of colored geometric objects is to be preprocessed so that given some query object, q, the distinct colors of the objects intersected by q can be reported efficiently or the number of such colors can be counted efficiently. In the dynamic setting, colored objects can be inserted into or deleted from S. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the techniques known for the standard problems do not yield efficient solutions for the generalized problems. Moreover, previous work (R. Janardan and M. Lopez, Generalized intersection searching problems, Internat. J. Comput. Geom. Appl.3 (1993), 39-69) on generalized problems applies only to the static reporting problems. In this paper, a uniform framework is presented to solve efficiently the counting/reporting versions of a variety of generalized intersection searching problems in static/dynamic settings. These problems include 1-, 2-, and 3-dimensional range searching, quadrant searching, interval intersection searching, 1- and 2-dimensional point enclosure searching, and orthogonal segment intersection searching.