Each categorization suffers from the same illness: the lack of resilience. The concept of textuality itself clearly distinguish between the ideal actors of writers and readers, laying on the well-established semiotic relational cluster of symbol-reference-referent, and frames the idea of text within some specific properties and norms. However, new unsettling forms of texts make us question the rigidity of those borders: namely, hypertexts completely reimagine the layout (the physical organization), the structure (the logical organization, the plot) and the semantics (the story) and the relations among these three features of a text. What makes hypertexts  so special is: i) their interactivity (freedom of acting with/within the text), ii) their multi-sequencing structure (being nonlinear, built up on a fluid hierarchy), iii) their freedom of composition (throughout associations and joints), iv) their browsing behaviour both in writing and reading, v) their procedural character (continuous and dynamic), vi) their openness (freedom of interpretation). All these features seem to place hypertext in the margins of textuality, beyond (hyper-) textuality but towards a new dimension still not completely understandable; however hypertexts still lay within this category, being coherent, cohesive, intentional, informative, etc. My aim is first to present hypertexts as “texts on the border”, analysing the aforementioned features (following the birth of the concept with Ted Nelson and Tim Berners-Lee), then to use hypertexts to unveil (some of) the “borders of texts” – i.e. the limits of a traditional idea of textuality. This will be done especially (but not only, as this distinction is now blurred) for the role of the reader, reinterpreting the phenomenology of the act of reading (as in Ingarden and Iser) within the frame of a deconstructed text (with Derrida, Deleuze, Barthes, etc).