In this paper the Subdomain Generation Method (SGM), originally formulated in Khan & Topping (1993; Khan, A. I. & Topping, B. H. V., Subdomain generation for parallel finite element analysis. Comput. Syst. Engng, 1993, 4(4/6), 473-488) for convex finite element domains, is generalized for arbitrary shaped domains. Modifications to the original SGM are described which allow partitioning of non-convex domains. These modifications have been made to the formulation of the optimization module and the predictive module. The examples presented in Khan & Topping (1993) have been re-worked and two more examples have been added which demonstrate the application of the method to arbitrary shaped domains. It is shown with the aid of the examples that the method provides well-balanced subdomains very efficiently and allows parallel adaptive mesh generation. The method in its present form may be used to partition unstructured graphs in two or three dimensions. Since the computational cost for the mesh partitioning with this method depends solely upon the initial coarse mesh, hence the computational cost does not increase with the increase in the mesh density of the final mesh. The method in its present form is unsuitable for relatively coarse grained parallel computers, however the modifications which would impart a greater degree of scalability to this method are discussed.