Variational formulation and piece-wise linear finite element approximations of Poisson's problem. Dirichlet and Neumann boundary conditions and Poincaré's and Friedrichs's inequalities. A word about linear elasticity. Condition numbers of finite element matrices and the preconditioned conjugate gradient method. Domains and subdomains. Subdomain matrices as building blocks for domain decomposition methods and the related Schur complements. The two-subdomain case: the Neumann--Dirichlet and Schwarz alternating algorithms; they can be placed in a unified framework and written in terms of Schur complements. Extension to the case of many subdomains; coloring, the problems of singular subdomain matrices, and the need to use a coarse, global problem. Three assumptions and the basic result on the condition number of additive Schwarz algorithms. Classical and more recent two--level additive Schwarz methods. Remarks on the effect of irregular subdomains. Extensions to elasticity problems including the almost incompressible case. Modern iterative substructuring methods: FETI–DP and BDDC. An introduction in terms of block-Cholesky for problems only partially assembled. The equivalence of the spectra. Results on elasticity including incompressible Stokes problems.