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 P-morphisms that was introduced by Hof, Knill, and Simon in 1995 [A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Comm. Math. Phys. 174 (1995) 149–159].