Author Name Disambiguation (AND) is a critical task for digital libraries aiming to link existing authors with their respective publications. Due to the lack of persistent identifiers used by researchers and the presence of intrinsic linguistic challenges, such as homonymy, the development of Deep Learning algorithms to address this issue has become widespread. Many AND deep learning methods have been developed, and surveys exist comparing the approaches in terms of techniques, complexity, performance. However, none explicitly addresses AND methods in the context of deep learning in the latest years (i.e. timeframe 2016-2024). In this paper, we provide a systematic review of state-of-the-art AND techniques based on deep learning, highlighting recent improvements, challenges, and open issues in the field. We find that DL methods have significantly impacted AND by enabling the integration of structured and unstructured data, and hybrid approaches effectively balance supervised and unsupervised learning.