Language assessment has a crucial role in the clinical diagnosis of several neurodegenerative diseases. The analysis of extended speech production is a precious source of information encompassing the phonetic, phonological, lexico-semantic, morpho-syntactic, and pragmatic levels of language organization. The knowledge about the distinctive linguistic variables identifying language deficits associated to different neurodegenerative diseases has progressively improved in the last years. However, the heterogeneity of such variables and of the way they are measured and classified limits any generalization and makes the comparison among studies difficult. Here we present an exhaustive review of the studies focusing on the linguistic variables derived from the analysis of connected speech samples, with the aim of characterizing the language disorders of the most prevalent neurodegenerative diseases, including primary progressive aphasia, Alzheimer's disease, movement disorders, and amyotrophic lateral sclerosis. A total of 61 studies have been included, considering only those reporting group analysis and comparisons with a group of healthy persons. This review first analyzes the differences in the tasks used to elicit connected speech, namely picture description, story narration, and interview, considering the possible different contributions to the assessment of different linguistic domains. This is followed by an analysis of the terminologies and of the methods of measurements of the variables, indicating the need for harmonization and standardization. The final section reviews the linguistic domains affected by each different neurodegenerative disease, indicating the variables most consistently impaired at each level and suggesting the key variables helping in the differential diagnosis among diseases. While a large amount of valuable information is already available, the review highlights the need of further work, including the development of automated methods, to take advantage of the richness of connected speech analysis for both research and clinical purposes.