We explain the induction process initiated by L. Zamboni and myself, which was designed to understand the word combinatorics of the natural codings, but is now better described through a geometrical model introduced by Delecroix and Ulcigrai, with a natural extension where convex polygons (parallelograms in the hyperelliptic case) replace the rectangles of the Rauzy-Veech induction. This induction is used to build families of examples of interval exchange transformations, with weak mixing or with eigenvalues, with Veech's simplicity property, or satisfying a criterion due to Bourgain which in turn implies Sarnak's conjecture on the orthogonality of the trajectories with the Moebius function.