In the last decade, bioprinting has emerged as a facile technique for fabricating tissues constructs mimicking the architectural complexity and compositional heterogeneity of native tissues. Amongst different bioprinting modalities, extrusion-based bioprinting (EBB) is the most widely used technique. Coaxial bioprinting, a type of EBB, enables fabrication of concentric cell-material layers and enlarges the scope of EBB to mimic several key aspects of native tissues. Over the period of development of bioprinting, tissue constructs integrated with vascular networks, have been one of the major achievements made possible largely by coaxial bioprinting. In this review, current advancements in biofabrication of constructs with coaxial bioprinting are discussed with a focus on different bioinks that are particularly suitable for this modality. This review also expounds the properties of different bioinks suitable for coaxial bioprinting and then analyses the key achievements made by the application of coaxial bioprinting in tissue engineering, drug delivery and in-vitro disease modelling. The major limitations and future perspectives on the critical factors that will determine the ultimate clinical translation of the versatile technique are also presented to the reader.