This talk provides a very general framework [1] that supplies a solid mathematical foundation of various meshless methods for solving time-independent PDE problems.At a glance,the basic steps are 1.Rewrite the PDE problem as a recovery problem [1,2] for an unknown function from infinitely many given data,e.g.a PDE and boundary values.2.Define rigorously what well-posedness means in this context.3.Define discretization as a computation of an unknown function from a finite-dimensional trial space from finitely many test data.The number of test data may exceed the dimension of the trial space("overtesting").4.Show that for all well-posed linear problems there is a sequence of discretizations that are uniformly stable and convergent,while using a sufficient amount of overtesting.5.Show that the error of such methods is completely determined by the error of approximation of the data of the true solution by the data of the trial space functions,measured in the norm defined via well-posedness of the problem.Spectral methods with spectral convergence are included.This also allows to compare convergence rates for weak and strong problems using the same trial space.6.Show how this applies to meshless methods including Trefftz techniques,MLPG,and unsymmetric collocation,and provide explicit convergence rates in Sobolev spaces.7.As a byproduct of the formulation as a recovery problem,symmetric collocation is shown to be an error-optimal method for problems in strong form [2].In particular,this means that all linear solvers for problems like e.g.Δu=f in Ω and u=g on Γ,if they are based on finitely many input values of f and g,will always be outperformed error-wise in a given Sobolev space by symmetric collocation using the kernel of that space.For weak problems,element-free Galerkin techniques are error-optimal.On the downside,this applies only to cases where the PDE data are exact evaluations of the PDE on trial or shape functions,thus excluding finite-difference,localized RBF or DMLPG [3] methods.To cover these techniques,and if time permits,it will be shown how to calculate worst-case relative error bounds in Sobolev spaces numerically.All of this will be illustrated by numerical examples.