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 healthcare 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 demand breakthroughs in almost all aspects of robotics, namely 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 sub-field. A particular focus of our paper lies in the discussions of these challenges and proposing future directions of research.