Beating of cilia and flagellae allows movement of the fluid surrounding isolated cells (for example: protists) or epithelia (bronchial tissue) but is also responsible for the movement of unicellular organisms in this medium (such as spermatozoa or protists). This paper aims to describe: (1) the biochemical and structural elements of the '9 + 2' structure called the axoneme; (2) the mechanisms of wave generation and propagation along the axoneme of cilia and flagellae are then described, stating that in most models of wave propagation, a clear distinction is made between the dynein-dependent microtubule sliding which represents the oscillatory motor and the bending mechanism which regulates wave propagation. In current models, the bending propagation is supported by a bind/relax cyclic mechanism which propagates in register, but frame-shifted, with the powering action of the dynein motor along the axoneme. While a large amount of knowledge was accumulated about the motor, little is known about the resisting elements regulating the bending. (3) The present study also puts forward ideas as to how these organelles have been highly conserved throughout eucaryotic evolution, and concludes with suggestions for further fields of investigation into this unique mechanical device used for cell movement.