The structure of the thesis is the following. The first chapter is an introduction to scanning microscopy, where the path that led to the Focused Ion Beam (FIB) is described and the main differences between electrons and ion beams are highlighted. Chapter 2 is what is normally referred to (which I do not really like) as ‘the theory chapter’. The theory of ion/matter interaction is presented in the first part of the chapter. The treatment is the standard one that can be found in the literature, but of course mine is the choice of the topics and the way in which they are presented. The second part of the chapter is a short introduction toMonte Carlo codes, and in particular to the two pieces of software that I have used for my basic simulations, SRIM/TRIM from J. Ziegler, and IONiSE from D. Joy. The third chapter is almost entirely made up of an article published in Microscopy & Microanalysis in 2011, on the subject of ion-induced Secondary Electron Emission; the paper is introduced by two small sections, the first being an introduction to ion beam imaging, the second presenting a standard theory of noise in scanning microscopy. Chapter 4 introduces the main topic of this thesis work, the one that is more dear to me: resolution in scanning ion microscopy. The chapter is a hybrid, for the theory of resolution that is presented is quite standard, but interlaced with a lot of rethinking and personal points of view, and finally adapted to the specific field of scanning ion microscopy. Chapters 5 and 6 are the first two articles published during my PhD, in Journal of Vacuum Science & Technology B. They both tackle the problem of resolution evaluation in scanning ion microscopy, the first for the Ga-FIB, the second for the He-FIB. The two papers have been published in 2008 and 2009, respectively, a time when the uniformity of the formalism was not yet mature, reason for which symbolic inconsistencies can be found in relation to the rest of the thesis, and to the list of symbols in appendix B. Choice was made to leave the papers in their published version, but this should not represent a problem, because each quantity is clearly defined. The last journal paper published within my project makes chapter 7. The scope of the paper is broad, for it proposes a method to simulate ion imaging that does not employ any Monte Carlo calculation. It can be regarded as a kind of summary of all the studies performed in the course of the project, and is followed by an appendix that explains the details of the noise analysis whose results are presented in the article. Conclusions and recommendations find their place in chapter 8. With referiment to the original main motivation, i.e. exploring the possibility of achieving atomic resolution with a Scanning IonMicroscope, it is shown that this is not possible, at least in the general case, because the sputtering of the sample limits the ultimate obtainable resolution to the nanometer range, even using very light ions. Also, it is pointed out that, in imaging systems causing strong sample modification, the concept of resolution itself can not be thought of as static; it must be regarded as dynamic instead. On the subject of Ion-Induced Secondary Electron Emission, a procedure to obtain curves of Secondary Electron yield versus incident angle of the beam is presented; the behaviour of those curves contributes to explain the much sharper contrast achievable with a He-FIB, as opposite to the more traditional Ga-FIB. Finally, it is shown that simulation of ion imaging based on the ‘yield vs. incidence angle’ curves is feasible, overcoming the computational problems that affect any Monte Carlo based approach.