  The aim of this review, based on a series of four lectures held at the 22nd
"Saalburg" Summer School (2016), is to cover selected topics in the theory of
perturbation series and their summation. The first part is devoted to
strategies for accelerating the rate of convergence of convergent series,
namely Richardson extrapolation, and the Shanks transformations, and also
covers a few techniques for accelerating the convergence of Fourier series. The
second part focuses on divergent series, and on the tools allowing one to
retrieve information from them. These techniques include Euler summation, Borel
summation, generic summation, and the method of continued functions, including
in particular the Pad\'e theory based on continued fractions. Finally, a brief
discussion of Stieltjes series and functions is given in order to study the
convergence properties of Pad\'e approximants.
