Defining the organisation of species interaction networks and unveiling the processes behind their assembly is fundamental to understanding patterns of biodiversity, community stability and ecosystem functioning. Marine sponges host complex communities of microorganisms that contribute to their health and survival, yet the mechanisms behind microbiome assembly are largely unknown. We present the global marine sponge-microbiome network and reveal a modular organisation in both community structure and function. Modules are linked by a few sponge species that share microbes with other species around the world. Further, we provide evidence that abiotic factors influence the structuring of the sponge microbiome when considering all microbes present, but biotic interactions drive the assembly of more intimately associated 'core' microorganisms. These findings suggest that both ecological and evolutionary processes are at play in host-microbe network assembly. We expect mechanisms behind microbiome assembly to be consistent across multicellular hosts throughout the tree of life.