Deformable object manipulation (DOM) is an emerging research problem in robotics. The ability to manipulate deformable objects endows robots with higher autonomy and promises new applications in the industrial, services, and health-care sectors. However, compared to rigid object manipulation, the manipulation of deformable objects is considerably more complex and is still an open research problem. Addressing DOM challenges demands breakthroughs in almost all aspects of robotics: hardware design, sensing, (deformation) modeling, planning, and control. In this article, we review recent advances and highlight the main challenges when considering deformation in each subfield. A particular focus of our article lies in the discussions of these challenges and proposing future directions of research.