Starting with the work of Hitoshi Nakada in the early 1980s, for both the metric and 
the arithmetic properties of various continued fraction expansion algorithms, 
the natural extension has played a key role in the last 40 years. In this talk, 
I review this, show how old results can be obtained in a short and elegant way, 
and also present some new results.
The new work presented is joint work with A. Bakhtawar.