This book originates from the lecture notes of the Ph.D. course on numerical simulations for strongly-correlated systems at the International School for Advanced Studies (S.I.S.S.A.) in Trieste. Even though the backbone of this book has been formed over the last fifteen years, almost all chapters have been completely rewritten, reshaped, or heavily modified in order to improve the clarity of the presentation. In addition, the second part of this book (i.e., Chapters 2, 3, and 4) has been taught during the C.E.C.A.M. Summer School on “Atomistic Simulation Techniques for Material Science, Nanotechnology, and Biophysics”, held in Trieste from 2011 to 2016. This school was addressed to graduate and undergraduate students who had little experience in numerical simulations and part of the actual presentation has benefited from their suggestions. The book is envisaged for students and young researches, who do not know much on Monte Carlo methods and want to understand how to implement ground-state algorithms on interacting models. After a first chapter dealing with a bird's-eye review on correlated wave functions that have been used in the past, the second part is intended to give a rather pedagogical introduction to simple probability theory (just the concepts that are necessary for the further developments) and methods for statistical samplings, not only based on Monte Carlo techniques but also including molecular-dynamics approaches. The central part of the book deals with variational and projection Monte Carlo approaches that are well established and widely used to treat lattice models (for both fermions and bosons). Finally, the last part provides an introduction to Monte Carlo methods that have been recently developed for electron systems on the continuum.

Our main motivation to convert the preliminary lecture notes into a structured book is to put in a sequential order all the concepts and machineries that are needed for writing efficient codes based upon variational and projection Monte Carlo techniques. Moreover, a modern textbook describing how to treat wave functions within quantum Monte Carlo algorithms was lacking in the portfolio of books treating correlated systems. The methods chosen here are influenced by our own research and do not cover all the numerical approaches that are currently used in the field.