Nijenhuis geometry

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