This paper focuses on a powerful and comprehensive overview of Deep Learning (DL) techniques on Distribution Automation System (DAS) applications to provide a complete viewpoint of modern power systems. DAS is a crucial approach to increasing the reliability, quality, and management of distribution networks. Due to the importance of development and sustainable security of DAS, the use of DL data-driven technology has grown significantly. DL techniques have blossomed rapidly, and have been widely applied in several fields of distribution systems. DL techniques are suitable for dynamic, decision-making, and uncertain environments such as DAS. This survey has provided a comprehensive review of the existing research into DL techniques on DAS applications, including fault detection and classification, load and energy forecasting, demand response, energy market forecasting, cyber security, network reconfiguration, and voltage control. Comparative results based on evaluation criteria are also addressed in this manuscript. According to the discussion and results of studies, the use and development of hybrid methods of DL with other methods to enhance and optimize the configuration of the techniques are highlighted. In all matters, hybrid structures accomplish better than single methods as hybrid approaches hold the benefit of several methods to construct a precise performance. Due to this, a new smart technique called Learning-to-learning (L2L) based DL is proposed that can enhance and improve the efficiency, reliability, and security of DAS. The proposed model follows several stages that link different DL algorithms to solve modern power system problems. To show the effectiveness and merit of the L2L based on the proposed framework, it has been tested on a modified reconfigurable IEEE 32 test system. This method has been implemented on several DAS applications that the results prove the decline of mean square errors by approximately 12% compared to conventional LSTM and GRU methods in terms of prediction fields.