In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of so-called rich words is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with X. Droubay and G. Pirillo in 2001. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of words.Amongst our main results, we give an explicit description of the periodic rich infinite words and show that the recurrent balanced rich infinite words coincide with the balanced episturmian words. We also consider two wider classes of infinite words, namely weakly rich words and almost rich words (both strictly contain all rich words, but neither one is contained in the other). In particular, we classify all recurrent balanced weakly rich words. As a consequence, we show that any such word on at least three letters is necessarily episturmian; hence weakly rich words obey Fraenkel's conjecture. Likewise, we prove that a certain class of almost rich words obeys Fraenkel's conjecture by showing that the recurrent balanced ones are episturmian or contain at least two distinct letters with the same frequency.Lastly, we study the action of morphisms on (almost) rich words with particular interest in morphisms that preserve (almost) richness. Such morphisms belong to the class of Pmorphisms that was introduced by A. Hof, O. Knill, and B. Simon in 1995.

We prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values, and begins with arbitrary long blocks which are ''almost squares,'' then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence (1+ (NnaM mod 2)) n \ 0 , or are a ''repetitive'' fixed point of a binary morphism satisfying some technical conditions.