INTRODUCTION Due to the work in [Brieskorn], [Tjurina], [Artin], [Wahl 2] and [Lipman] one understands very well the notion of simultaneous resolution for rational surface singularities. Given a rational surface singularity X, there exists a smooth parameter space Res representing the functor of deformations of the (minimal) resolution. The family Y ---+ Res contracts to X ---+Res. The fibers yt---+ X 1 are the minimal resolutions of Xt. There is a finite and Galois map Res ---+ Defx (the versa! base space of X). The image is the Artin component, which represents the functor of deformations of X with simultaneous resolution after finite base change, and the covering group W is a reflection group which is also the monodromy group of the component. In the case of rational double points (RDPs) W is the Weyl group of the corresponding root system (Ak, Dk, E6, E1 orEs). In general much information about the deformation, e.g. the discriminant and adjacencies, may be read off the geometry of this covering. The reflections in W are related to certain divisors, called roots, on Y and their liftings. The contraction map is induced by contracting RDP-configurations on Y. In connection with his work on deformations of cyclic quotient singularities ( CQS), the second author observed a natural Galois covering of each component in the reduced versal base space of a CQS. In the case of the Artin component the observed covering group was the same as for Res---+ Def. (See [Christophersen 1] and [Christophersen 2].) The first author conjectured that this was the monodromy cover and we asked ourselves if there was a deformation theoretic explanation. The purpose of this paper is to answer this question. We will show that something similar to the Res ---+ Def picture actually happens for every non-embedded component of the versa! base space of a quotient singularity. (For us a quotient singularity is the singularity of C2 /G where G C GL(2, C) is a finite subgroup which we can assume to be without pseudo-reflections.) For quotient singularities, the construction of the Artin component is a special case of a procedure involving deformations of certain modifications which we call M-resolutions. The application of threefold theory to deformations of rational singularities as found in [Kollfu--Shepherd-Barron] was important both for discovering M-resolutions and for the proofs of their properties. In fact, our results are anticipated in Theorem 3.5(a) of that paper. Still the definitions and statement of the main result may be made without reference to that work. The relevant results from [Kollfu--Shepherd-Barron] are postponed to §1. Consider a smoothing of a normal surface singularity X with smooth generic fiber F. The Milnor number of this smoothing is p. = rk H2 (F) and depends upon the component of the versa! base space of X on which the smoothing appears.