This book is based on several courses taught by the author at the University of Michigan between 1908 and 1912. It covers two main topics: asymptotic series and the theory of summability. The discussion of nowhere convergent asymptotic series includes the so-called MacLaurent summation formula, determining asymptotic expansions of various classes of functions, and the study of asymptotic solutions of linear ordinary differential equations. On the second topic, the author discusses various approaches to the summability of divergent series and considers in detail applications to Fourier series.