Natural hydrogels are one of the most promising biomaterials for tissue engineering applications, due to their biocompatibility, biodegradability, and extracellular matrix mimicking ability. To surpass the limitations of conventional fabrication techniques and to recapitulate the complex architecture of native tissue structure, natural hydrogels are being constructed using novel biofabrication strategies, such as textile techniques and three-dimensional bioprinting. These innovative techniques play an enormous role in the development of advanced scaffolds for various tissue engineering applications. The progress, advantages, and shortcomings of the emerging biofabrication techniques are highlighted in this review. Additionally, the novel applications of biofabricated natural hydrogels in cardiac, neural, and bone tissue engineering are discussed as well.