In this thesis, we present an expanded account of domain decomposition for solving PDEs using RBF collocation methods based on two articles by Chinchapatnam and coworkers (2006) and Mai-Duy et al. (2008). The well-known finite difference, finite element and finite volume methods in solving partial differential equations are based on a mesh discretization which is a complicated and time consuming process particularly for complex higher dimensional geometries. The meshfree or meshless methods try to circumvent the cumbersome issues of mesh generation. One of the meshless methods is due to the pioneering effort of Kansa (1990) who directly collocated the RBFs for the approximate solutions of differential equations. Kansa method which is known as DRBF (Differentiated RBF) collocation method, has several advantages in comparison with traditional methods, and has been applied successfully to obtain numerical solution of various type of ordinary and partial differential equations. Also Mai-Duy, Tran-Cong presented similar method which is known as IRBF (Integrated RBF) collocation method (1999). RBF collocation methods are very simple to implement because they are truly meshless in the sense of that collocation points need not have any connectivity requirement as needed traditional methods. They are spatial dimension independent which is very attractive for modeling high dimensional problems. They possess superior rate of convergence too. Therefore, for small to moderate sized problems, RBF collocation methods do outperform traditional methods but for large scale problems, the resultant coefficient matrix is highly ill-conditioned, which hinders the applicability of the RBF collocation methods. One of the best remedies to ill-conditioning problem is domain decomposition, which has presented by Dubal for DRBF (1994), and Mai-Duy et al. for IRBF (2008). We first study the interpolation by radial basis functions, which is used to constructing RBF collocation methods. We then offer the DRBF and IRBF collocation methods for solving ODEs and PDEs. After that we present different type of DD and combine them with RBF collocation methods. Finally we show the efficiency of the presented methods by various kinds of numerical examples. 2000 MSC: 65M55 Key words : Radial Basis Function (RBF), RBF collocation methods, Domain Decomposition (DD).