This book presents an alternative to what the topologists refer to as “Morse–Novikov theory”, a mathematical theory which belongs to the fields of geometry and topology. The theory presented has interest in topology and dynamics, has provided inspiration and has applications outside of mathematics, especially in data analysis and shape recognitions in physics and computer science.It describes a new class of invariants associated with a generic continuous real or angle valued map defined on a compact metrizable space based on homology with coefficients in a given field. The invariants are finite, computable by implementable algorithms in case the underlying space of the map has a triangulation and the map is simplicial, and are, in some sense, the analogues of the set of trajectories between rest points and of closed trajectories of a generic vector field (which admits a Lyapunov closed one form) on a smooth manifold. These can be used to conclude existence of such trajectories even for flows on compact metric spaces which are not smooth and cannot be described via differential calculus. Two alternative definitions of these invariants based on different mathematics and algorithms for their calculation will be described. The book presents remarkable properties (stability, Poincaré-duality) of these invariants and will relate them to the global algebraic topology of the space the map is defined on, in the spirit of Morse–Novikov theory.