DNAs exist in nature and could provide solutions to computing, storage and communications as existing ones approach their physical limits. Plenty of research work has been conducted on DNA-based computing, data storage and molecular communications, but largely in isolation. There is a lack of a unified place where the triples are put together to be discussed. This paper aims to filling in this gap by providing an overview of each triplet. Starting from an overall description of DNA features and their reading and writing in practical terms, the paper goes on to describe each of the three from three aspects: requirements and differences from current electronic-dominating technologies, their working principles and practical considerations. Recent advancement in each area is summarized and discussed. Furthermore, the paper intends to call for researches that go beyond the boundary of each and encourages interconnection and joint research among the three. It proposes a molecular ICT system architecture with all its three components underpinned by DNAs. The paper also identifies and discusses some new future directions such as joint coding for storage and communications, directional DNA-based molecular communications, interfaces between molecular DNA systems and electronic systems. It is hoped that this paper can spark more joint research across computing, storage and communications in this exciting field of DNA-based molecular ICT systems.