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Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely employed to solve partial differential equations with discontinuous solutions. However, stable ENO/WENO methods on unstructured grids are less well studied. We propose high-order essentially nonoscillatory methods based on radial basis functions (RBFs) to solve hyperbolic conservation laws. We derive a smoothness indicator that guarantees the satisfaction of the sign property of the resulting interpolant on general one-dimensional grids. Based on this algorithm we introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Hence, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. Next, we propose a high-order ENO method based on radial basis functions to solve hyperbolic conservation laws on general two-dimensional unstructured grids. The radial basis function reconstruction offers a flexible framework to deal with ill-conditioned cell arrangements. We define a smoothness indicator based the one-dimensional version and a stencil selection algorithm suitable for general meshes. Furthermore, we develop a stable method to evaluate the RBF reconstruction in the finite volume setting to circumvent the stagnation of the error and keep the condition number of the reconstruction bounded. To reduce the computational complexity, we develop the RBF-CWENO method. This method exhibits high-order convergence and robustness when solving challenging problems and is considerably faster. However, the resolution close to shocks and turbulent structures is lower than for the RBF-ENO method. Finally, we present a hybrid high-resolution RBF-ENO method which is based on the RBF-ENO method for unstructured patches and the standard WENO method on structured ones. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods that stabilizes the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness on the scramjet inflow problem and a conical aerospike nozzle jet simulation.