Domain decomposition, a form of divide-and-conquer for mathematical problems posed over a physical domain is the most common paradigm for large-scale simulation on massively parallel, distributed, hierarchical memory computers. In domain decomposition, a large problem is reduced to a collection of smaller problems, each of which is easier to solve computationally than the undecomposed problem, and most or all of which can be solved independently and concurrently. Domain decomposition has proved to be an ideal paradigm not only for execution on advanced architecture computers, but also for the development of reusable, portable software. The most complex operation in a typical domain decomposition method – the application of the preconditioner – carries out in each subdomain steps nearly identical to those required to apply a conventional preconditioner to the undecomposed domain. Hence software developed for the global problem can readily be adapted to the local problem, instantly presenting lots of legacy scientific code for to be harvested for parallel implementations. Finally, it should be noted that domain decomposition is often a natural paradigm for the modeling community. Physical systems are often decomposed into two or more contiguous subdomains based on phenomenological considerations, and the subdomains are discretized accordingly, as independent tasks. This physically-based domain decomposition may be mirrored in the software engineering of the corresponding code, and leads to threads of execution that operate on contiguous subdomain blocks. This tutorial provides an overview of domain decomposition and focuses on the mathematical development of its two main paradigms: Schwarz and Schur preconditioning and their hybrids.