Background: In the past few years there has been a growing interest in the employment of verbal productions as digital biomarkers, namely objective, quantifiable behavioural data that can be collected and measured by means of digital devices, allowing for a low-cost pathology detection, classification and monitoring. Numerous research papers have been published on the automatic detection of subtle verbal alteration, starting from written texts, raw speech recordings and transcripts, and such linguistic analysis has been singled out as a cost-effective method for diagnosing dementia and other medical conditions common among elderly patients (e.g., cognitive dysfunctions associated with metabolic disorders, dysarthria). Aims: To provide a critical appraisal and synthesis of evidence concerning the application of natural language processing (NLP) techniques for clinical purposes in the geriatric population. In particular, we discuss the state of the art on studying language in healthy and pathological ageing, focusing on the latest research efforts to build non-intrusive language-based tools for the early identification of cognitive frailty due to dementia. We also discuss some challenges and open problems raised by this approach. Methods & Procedures: We performed a scoping review to examine emerging evidence about this novel domain. Potentially relevant studies published up to November 2021 were identified from the databases of MEDLINE, Cochrane and Web of Science. We also browsed the proceedings of leading international conferences (e.g., ACL, COLING, Interspeech, LREC) from 2017 to 2021, and checked the reference lists of relevant studies and reviews. Main Contribution: The paper provides an introductory, but complete, overview of the application of NLP techniques for studying language disruption due to dementia. We also suggest that this technique can be fruitfully applied to other medical conditions (e.g., cognitive dysfunctions associated with dysarthria, cerebrovascular disease and mood disorders). Conclusions & Implications: Despite several critical points need to be addressed by the scientific community, a growing body of empirical evidence shows that NLP techniques can represent a promising tool for studying language changes in pathological aging, with a high potential to lead a significant shift in clinical practice.