The first problems in this Chapter present a survey of the basic properties of analytic functions of a complex variable, of their powers series expansions (Taylor-Laurent series), of their singularities, including branch points and cut lines. As an application, the evaluation of many types of integrals by complex variable methods is proposed. Some examples of conformal mappings are finally studied, mainly in order to find harmonic functions and to solve Dirichlet Problems; the results are compared with those obtained in other Chapters with different methods, with a discussion about the uniqueness of the solutions.