Over the last decade, 3D bioprinting has received immense attention from research communities for developing functional tissues. Thanks to the complexity of tissues, various bioprinting methods have been exploited to figure out the challenges of tissue fabrication, in which hydrogels are widely adopted as a bioink in cell printing technologies based on the extrusion principle. Thus far, there is a wealth of literature proposing the crucial parameters of extrusion-based bioprinting of hydrogel biomaterials (e.g., hydrogel properties, printing conditions, and tissue scaffold design) toward enhancing performance. Despite the growing research in this field, numerous challenges that hinder advanced applications still exist. Herein, the most recently reported hydrogel-based bioprinted scaffolds, i.e., skin, bone, cartilage, vascular, neural, and muscular (including skeletal, cardiac, and smooth) scaffolds, are systematically discussed with an emphasis on the advanced fabrication techniques from the tissue engineering perspective. The methods covered include multiple-dispenser, coaxial, and hybrid 3D bioprinting. The present work is a unique study to figure out the opportunities of the novel techniques to fabricate complicated constructs with structural and functional heterogeneity. Finally, the principal challenges of current studies and a vision of future research are presented.