Combinations of complex dynamical systems This work is a research level monograph whose goal is to develop a general combination decomposition and structure theory for branched coverings of the two sphere to itself regarded as the combinatorial and topological objects which arise in the classification of certain holomorphic dynamical systems on the Riemann sphere. It is intended for researchers interested in the classification of those complex one dimensional dynamical systems which arein some loose sensetame. The program is motivated by the dictionary between the theories of iterated rational maps and Kleinian groups. TOC:Introduction. Preliminaries. Combinations. Uniqueness of combinations. Decompositions. Uniqueness of decompositions. Counting classes of annulus maps. Applications to mapping class groups.Examples. Canonical decomposition theorem