  Generalized cycles can be thought of as the extension of form-cycle duality
between holomorphic forms and cycles, to meromorphic forms and generalized
cycles. They appeared as an ubiquitous tool in the study of spectral curves and
integrable systems in the topological recursion approach. They parametrize
deformations, implementing the special geometry, where moduli are periods, and
derivatives with respect to moduli are other periods, or more generally
"integrals", whence the name "generalized cycles". They appeared over the years
in various works, each time in specific applied frameworks, and here we provide
a comprehensive self-contained corpus of definitions and properties for a very
general setting. The geometry of generalized cycles is also fascinating by
itself.
