This paper presents an initial experimental and numerical study on a novel concept of integrated tensegrity/origami morphing structures. Both tensegrity and origami are known for their potential transformative roles in applications in fields which exist at the interplay of shape, size, and function. Their integration, proposed here for the first time, is based on the interest in uniting the strengths of origami (membrane integration and folding capabilities) with the strengths of tensegrity (minimal mass and controllable rigidity). In order to achieve morphing capabilities while retaining low mass, the considered structures possess intrinsic material actuation provided by shape memory alloy (SMA) members. Two different representative structures of the proposed concept are studied. The first corresponds to a tensegrity/origami cylinder based on the double-helix tensegrity topology. Shape memory alloy wires play the role of the tensile members of the tensegrity structure while foldable surface components play the role of compressive members. The second structure corresponds to a tensegrity plate, which although not having a continuous dense surface, can be morphed in novel ways using origami principles. Fabrication of experimental prototypes and evaluation of the observed structural transformation are presented. Finite element analysis is performed to numerically evaluate the characteristics of the novel proposed structures. This initial study shows that the integration of tensegrity and origami results in a powerful combination that provides structural stability, flexibility in design and lightweight actuation mechanisms while allowing for significant deflections during morphing.