The aim of this talk is to highlight some interesting specimens in the jungle of continued fraction algorithms. Our view of continued fractions is an algorithmic one, giving an enormous choice of possible expansions. Finite state automata can be used to characterize digit sequences that will arise for certain choices. Details in this area tend to become very technical, but most ideas involved are elementary, so without being too superficial we will give explanations that will be pleasing to eye, ear and brain alike.

We will discuss several well-known results on sums of bounded continued fractions, on algebraicity for bounded digits, and on recognizability by finite automata. We present generalizations to the complex case; some of these results have not appeared yet.