The purpose of these lectures is to clarify such subjects as inverse problems, inverse methods, data assimilation, reanalyses, state estimation, etc. Although these subjects can be presented in very interesting mathematical contexts, in their practical use on computers the ideas of elementary linear algebra, finite dimensional vector spaces, and basic statistics suffice for an understanding of what they try to do and how they do it. I will attempt to separate the challenging problems of numerical engineering (how to make things work in practice for very large models and data sets) from the...