Three-dimensional (3D) bioprintingis a well-known promising technology for the production of artificial biological organs providing unprecedented versatility for manipulating cells and other biomaterials with precise control of their location in space. Over the past decade, a number of 3D bioprinting technologies have been developed. Unlike traditional manufacturing technologies, 3D bioprinting allows to produce individual or personalized fabric designs. This helps to deposit cells of the desired type with selected biomaterials and desired biologically active substances. Natural polymers play a leading role in maintaining cellular and biomolecular processes before, during, as well as after three-dimensional bioprinting. Polymers of biological origin can be extracted from natural raw materials by means of physical or chemical methods. These polymers are widely used as effective hydrogels for loading cells to form tissues, build a vascular, nervous, lymphatic network, and also to implement multiple biological, biochemical, physiological, biomedical and other functions. Any natural polymers that have a sol-gel phase transition (i.e., a gel point) under certain conditions can be printed using the automatic layer-by-layer deposition method. In fact, very few of them can be printed under various conditions (low temperature, without the help of physical, chemical, biochemical crosslinking of the incorporated polymer chains). Thus, not all natural polymers can meet all the basic requirements for 3D bioprinting. As a rule, natural polymers as the main component of various inks, which contain cells suspended in a specific medium, must meet several basic requirements for successful 3D bioprinting of organs, as well as clinical applications. These include biocompatibility, that is, non-toxic or without apparent toxicity; biodegradability (unlikenon-biodegradable polymers can be used as auxiliary structures); biostability with sufficiently high mechanical strength both at the time of processing and during operation; bioprinterness (workability). This review is devoted to modern research in the field of natural polymers used to print biological artificial organs.