Given an infinite word over the alphabet {0, 1, 2, 3}, we define a class of bipartite hereditary graphs G α , and show that G α has unbounded clique-width unless α contains at most finitely many non-zero letters.We also show that G α is minimal of unbounded clique-width if and only if α belongs to a precisely defined collection of words Γ . The set Γ includes all almost periodic words containing at least one non-zero letter, which both enables us to exhibit uncountably many pairwise distinct minimal classes of unbounded clique width, and also proves one direction of a conjecture due to Collins, Foniok, Korpelainen, Lozin and Zamaraev. Finally, we show that the other direction of the conjecture is false, since Γ also contains words that are not almost periodic.of unbounded treewidth (see Robertson and Seymour [17]), and circle graphs are the unique minimal vertex-minor-closed class of unbounded rank-width (or, equivalently, clique-width)see Geelen, Kwon, McCarty and Wollan [11].The situation for clique-width and hereditary classes is much more complicated, yet remains of significant interest (note that every vertex-minor-closed class is a hereditary class, but not vice-versa). First, there exist hereditary classes of graphs that have unbounded clique-width, but which contain no minimal class of unbounded clique-width: it is well-known that the class of square grid graphs has this property; a more recent example is due to Korpelainen [13], who also suggests possible ways to handle such classes.However, there do also exist minimal hereditary classes of unbounded clique-width. We refer the reader to the excellent survey by Dabrowksi, Johnson and Paulusma [8] for further details of the progress in recent years. Of particular relevance here is the work of Collins, Foniok, Korpelainen and Lozin [3], in which a countably infinite family of minimal hereditary classes of unbounded clique-width is given.More precisely, the authors of [3] construct hereditary bipartite graph classes by taking the finite induced subgraphs of an infinite graph whose vertices form a two-dimensional array and whose edges are defined by an infinite word over the alphabet {0, 1, 2}. They show that classes defined by an infinite periodic word over the alphabet {0, 1} are minimal of unbounded cliquewidth, and conjecture that a class defined by a word over the alphabet {0, 1, 2} is minimal of unbounded clique-width if and only if the word is almost periodic, and not the all-zeros word.In this paper, we go further by proving the following results.Theorem 1.1. Let α be an infinite word over the alphabet {0, 1, 2, 3}, and let G α be the corresponding hereditary graph class (as defined in Section 2).(a) If α has an infinite number of non-zero letters, then G α has unbounded clique-width (Theorem 3.10).(b) If α is an almost periodic word with at least one non-zero letter then G α is minimal of unbounded clique-width (Theorem 4.9).(c) The number of distinct minimal hereditary classes of graphs of unbounded clique-width is uncountably infinite (Theorem 4.11...