This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such as *Stochastic Networks*, *Analytic Combinatorics*, and *Quantum Physics*. This second edition consists of two parts.

Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By using *Complex Function Theory*, *Boundary* *Value Problems*, *Riemann Surfaces*, and *Galois Theory*, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize the *Transient Behavior* of the walks, as well as to find explicit solutions to the one-dimensional *Quantum Three-Body Problem*, or to tackle a new class of Integrable Systems.

**Part II** borrows special case-studies from queueing theory (in particular, the famous problem of *Joining the Shorter of Two Queues*) and enumerative combinatorics (*Counting*, *Asymptotics*).

Researchers and graduate students should find this book very useful.