A mesh-free method for solving linear and nonlinear multi-dimensional transport problems is proposed. This method is essentially a combination of an adaptive semi-Lagrangian method with a mesh-free radial basis function interpolation on scattered data. We discuss customized strategies for node placement for achieving locally optimally spaced node clusters for the approximation of PDEs. A special a posteriori error indicator is developed, in order to refine/coarsen the nodes according to approximation requirements. Some remarks on the stability of the proposed method are made. Numerical examples in two space dimensions demonstrate the effectiveness of the meshfree method of backward characteristics. A model problem and a near-realistic problem for linear transport are treated yielding accurate results. The new method proves to be applicable to nonlinear equations. Examples of Burger’s equation and Buckley-Leverret equation in two dimensions are given. An artificial viscosity ansatz is chosen to model shock propagation with the characteristics based method.