Polyurethanes (PUs) are a significant group of polymeric materials that, due to their outstanding mechanical, chemical, and physical properties, are used in a wide range of applications. Conventionally, PUs are obtained in polyaddition reactions between diisocyanates and polyols. Due to the toxicity of isocyanate raw materials and their synthesis method utilizing phosgene, new cleaner synthetic routes for polyurethanes without using isocyanates have attracted increasing attention in recent years. Among different attempts to replace the conventional process, polyaddition of cyclic carbonates (CCs) and polyfunctional amines seems to be the most promising way to obtain non-isocyanate polyurethanes (NIPUs) or, more precisely, polyhydroxyurethanes (PHUs), while primary and secondary -OH groups are being formed alongside urethane linkages. Such an approach eliminates hazardous chemical compounds from the synthesis and leads to the fabrication of polymeric materials with unique and tunable properties. The main advantages include better chemical, mechanical, and thermal resistance, and the process itself is invulnerable to moisture, which is an essential technological feature. NIPUs can be modified via copolymerization or used as matrices to fabricate polymer composites with different additives, similar to their conventional counterparts. Hence, non-isocyanate polyurethanes are a new class of environmentally friendly polymeric materials. Many papers on the matter above have been published, including both original research and extensive reviews. However, they do not provide collected information on NIPU composites fabrication and processing. Hence, this review describes the latest progress in non-isocyanate polyurethane synthesis, modification, and finally processing. While focusing primarily on the carbonate/amine route, methods of obtaining NIPU are described, and their properties are presented. Ways of incorporating various compounds into NIPU matrices are characterized by the role of PHU materials in copolymeric materials or as an additive. Finally, diverse processing methods of non-isocyanate polyurethanes are presented, including electrospinning or 3D printing.