3D printed hydrogels have emerged as a novel tissue engineering and regeneration platform due to their ability to provide a suitable environment for cell growth. To obtain a well-defined scaffold with good post-printing shape fidelity, a proper hydrogel ink formulation plays a crucial role. In this regard, alginate has received booming interest owing to its biocompatibility, biodegradability, easy functionalization, and fast gelling behavior. Hence, this review highlights the significance of alginate-based hydrogel inks for fabricating 3D printed scaffolds for bone and cartilage regeneration. Herein, we discuss the fundamentals of direct extrusion 3D bioprinting method and provide a comprehensive overview of various alginate-based hydrogel ink formulations that have been used so far. We also summarize the requirements of hydrogel inks and 3D printed scaffolds to achieve similarity to the native tissue environment. Finally, we discuss the challenges, and research directions relevant for future clinical translation.