What is spectral action how to compute it and what are the known examples? This book offers a
guided tour through the mathematical habitat of noncommutative geometry à la Connes
deliberately unveiling the answers to these questions. After a brief preface flashing the
panorama of the spectral approach a concise primer on spectral triples is given. Chapter 2 is
designed to serve as a toolkit for computations. The third chapter offers an in-depth view into
the subtle links between the asymptotic expansions of traces of heat operators and meromorphic
extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the
spectral action under fluctuations by gauge potentials. A subjective list of open problems in
the field is spelled out in the fifth Chapter. The book concludes with an appendix including
some auxiliary tools from geometry and analysis along with examples of spectral geometries.
The book serves both as a compendium for researchers in the domain of noncommutative geometry
and an invitation to mathematical physicists looking for new concepts.
