  This work is the first, and main, of the series of papers in progress
dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing
Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that
should be understood in much wider context than before: from local description
at generic points to singularities and global analysis. The goal of the present
paper is to introduce terminology, develop new important techniques (e.g.,
analytic functions of Nijenhuis operators, splitting theorem and
linearisation), summarise and generalise basic facts (some of which are already
known but we give new self-contained proofs), and more importantly, to
demonstrate that the research programme proposed in the paper is realistic by
proving a series of new, not at all obvious, results.
