During the past two decades, tissue engineering and the regenerative medicine field have invested in the regeneration and reconstruction of pathologically altered tissues, such as cartilage, bone, skin, heart valves, nerves and tendons, and many others. The 3D structured scaffolds and hydrogels alone or combined with bioactive molecules or genes and cells are able to guide the development of functional engineered tissues, and provide mechanical support during in vivo implantation. Naturally derived and synthetic polymers, bioresorbable inorganic materials, and respective hybrids, and decellularized tissue have been considered as scaffolding biomaterials, owing to their boosted structural, mechanical, and biological properties. A diversity of biomaterials, current treatment strategies, and emergent technologies used for 3D scaffolds and hydrogel processing, and the tissue-specific considerations for scaffolding for Tissue engineering (TE) purposes are herein highlighted and discussed in depth. The newest procedures focusing on the 3D behavior and multi-cellular interactions of native tissues for further use for in vitro model processing are also outlined. Completed and ongoing preclinical research trials for TE applications using scaffolds and hydrogels, challenges, and future prospects of research in the regenerative medicine field are also presented.