This thesis advances the meshless Sequentially Optimized Meshfree Approximation (SOMA) from a fixed grid to an adaptive one by applying residual-based adaptive techniques. In its fixed grid form, SOMA constructs an approximation of an equation solution using optimized radial basis functions (RBFs), but deletes the RBF parameters once each basis function is appropriately added. The first proposed method saves this information, constructs an approximation of the solution, and intelligently adds points to the problem domain. The second proposed method is a flexible interpolation scheme which does not require this basis saving technique, although the two techniques can be combined. When applied to various equations, these adaptive algorithms demonstrate the convergence required to achieve a satisfactory level of precision, saving time and computational effort for the same mathematical result as a denser grid. Applications of this algorithm include function approximation as well as differential equations which demonstrate its capability and robustness.