The Thue-Morse word over the alphabet {a, b} is the fixed point starting with a of the morphism sending a to ab and b to ba, that is, it is the limit of the sequence of finite words a, ab, abba, abbabaab, abbabaabbaababba, … We survey the Diophantine properties of real numbers whose expansion in some integer base or whose continued fraction expansion is given by a Thue-Morse word (here, a and b are distinct positive integers). We also discuss the Diophantine properties of p-adic numbers whose Hensel expansion is a Thue-Morse word and of power series over a finite field F whose continued fraction expansion is a Thue-Morse word (here, a and b are distinct nonconstant polynomials with coefficients in F).