Background/Objectives: The rapid advancement of artificial intelligence (AI) has had a notable impact in the healthcare field, particularly in the realm of assessment and diagnosis. One specific area where the integration of AI technologies shows promise is the evaluation of progressive neurological disorders (PNDs). PNDs are characterized by a progressive decline in neurological function, resulting in changes in cognition, movement, and communication. PNDs pose significant challenges in terms of early detection and categorization. Speech and voice changes are important clinical markers in many PNDs. Therefore, the utilization of AI applications for the analysis and classification of speech and voice samples could prove beneficial for streamlining the diagnostic process. This systematic review aimed to investigate the current utilization of AI in the assessment and diagnosis of PNDs through speech signal analysis over the past decade. Methods: In adherence to PRISMA guidelines, Scopus, PubMed, and Web of Science were searched for studies related to machine learning (ML) and deep learning (DL) for speech and voice assessment in people with PNDs. Results: A total of 102 studies were identified for inclusion between 2013 and 2023. The reviewed studies demonstrated a wide range of accuracy, with reported values ranging from 67.43% to 99%. Support Vector Machines (SVMs) were the most frequently used ML models across studies, demonstrating reliable performance in both speech and voice data analysis. Conclusions: AI-based analysis of speech and voice shows strong potential as a non-invasive tool for supporting the assessment and diagnosis of PNDs. The high accuracy reported across studies highlights the promise of these approaches, although methodological variability underscores the need for greater standardization and clinical validation.