" In this article I will address the issue of the embodiment of computing sys-tems from the point of view distinctive of the so-called Unconventional Computation, focusing on the paradigm known as Mor-phological Computation. As a first step, I will contextualize Morphological Computa-tion within the disciplinary field of Embod-ied Artificial Intelligence: broadly con-ceived, Embodied Artificial Intelligence may be characterized as embracing both conventional and unconventional ap-proaches to the artificial emulation of natu-ral intelligence. Morphological Computa-tion stands out from other paradigms of unconventional Embodied Artificial Intelli-gence in that it discloses a new, closer kind of connection between embodiment and computation. I will further my investigation by briefly reviewing the state-of-the-art in Morphological Computation: attention will be given to a very recent trend, whose core concept is that of “organic reconfigu-rability”. In this direction, as a final step, two advanced cases of study of organic or living morphological computers will be pre-sented and discussed. The prospect is to shed some light on our title question: what progress has been made in understanding the embodiment of computing systems? Keywords: Embodied Artificial Intelligence; Morphological Computation; Reservoir Compu-ting Systems; Organic Reconfigurability; 3D Bio-Printed Synthetic Corneas; Xenobots "