ABSTRACT
In this chapter we discuss some properties of Hamel bases of the real line R and show their remarkable role in various constructions of strange additive functions acting from R into R.
The existence of such a basis was first established by Hamel in 1905 (see [70]). Later, it was also shown that the existence of a Hamel basis cannot be proved without the aid of uncountable forms of the Axiom of Choice (in particular, it is impossible to establish the existence of such a basis within the theory ZF & DC).
