The choice of material, manufacturing process, and molding tool significantly affects the quality, environmental impact, and cost efficiency of composite components. Producing one-piece hollow profiles with smooth inner surfaces and undercuts presents major challenges for conventional mold concepts. There is yet no thorough review of shape-variable mandrels in composite manufacturing to be found in the literature. This paper provides an overview of research on shape memory polymers and other shape-variable materials used in tooling applications for composite manufacturing. This work covers shape memory, heat shrink, and other deformable tooling concepts that enable the production of one-piece Type V pressure vessels, air intake ducts, or curved struts and tubes. A systematic literature review in combination with a state-of-the-art open-source active learning tool ASReview is conducted. Fifteen relevant studies were identified. Research on shape-variable tooling is mainly conducted by three research groups in the USA and the PRC. The tooling is mostly made of unreinforced thermosets, especially styrene-based ones. Thermoplastic resins are less common, and reinforcements limit the usable elongation in the temporary shape. The shape variability is either a shape memory and/or a softening process, which, in all studies, is activated by heating. Release agents are widely used to ease demolding. No ecological or economical assessment of the manufacturing methods was conducted in the reviewed studies. Three fields for further research that could be identified are as follows: (1) thorough ecological end economical assessment of shape-variable mandrels in comparison with conventional tooling; (2) thermoplastic shape memory polymer mandrels; and (3) further investigation of simulation capabilities for shape memory mandrels.