The aim of this paper is to provide a theoretical framework to understand how multicellular systems realize functionally integrated physiological entities by organizing their intercellular space. From a perspective centered on physiology and integration, biological systems are often characterized as organized in such a way that they realize metabolic self-production and self-maintenance. The existence and activity of their components rely on the network they realize and on the continuous management of the exchange of matter and energy with their environment. One of the virtues of the organismic approach focused on organization is that it can provide an understanding of how biological systems are functionally integrated into coherent wholes. Organismic frameworks have been primarily developed by focusing on unicellular life. Multicellularity, however, presents additional challenges to our understanding of biological systems, related to how cells are capable to live together in higher-order entities, in such a way that some of their features and behaviors are constrained and controlled by the system they realize. Whereas most accounts of multicellularity focus on cell differentiation and increase in size as the main elements to understand biological systems at this level of organization, we argue that these factors are insufficient to provide an understanding of how cells are physically and functionally integrated in a coherent system. In this paper, we provide a new theoretical framework to understand multicellularity, capable to overcome these issues. Our thesis is that one of the fundamental theoretical principles to understand multicellularity, which is missing or underdeveloped in current accounts, is the functional organization of the intercellular space. In our view, the capability to be organized in space plays a central role in this context, as it enables (and allows to exploit all the implications of) cell differentiation and increase in size, and even specialized functions such as immunity. We argue that the extracellular matrix plays a crucial active role in this respect, as an evolutionary ancient and specific (non-cellular) control subsystem that contributes as a key actor to the functional specification of the multicellular space and to modulate cell fate and behavior. We also analyze how multicellular systems exert control upon internal movement and communication. Finally, we show how the organization of space is involved in some of the failures of multicellular organization, such as aging and cancer.