High-throughput sequencing technologies produce large collections of data, mainly DNA sequences with additional information, requiring the design of efficient and effective methodologies for both their compression and storage. In this context, we first provide a classification of the main techniques that have been proposed, according to three specific research directions that have emerged from the literature and, for each, we provide an overview of the current techniques. Finally, to make this review useful to researchers and technicians applying the existing software and tools, we include a synopsis of the main characteristics of the described approaches, including details on their implementation and availability. Performance of the various methods is also highlighted, although the state of the art does not lend itself to a consistent and coherent comparison among all the methods presented here. 