Computational simulations are a popular method for testing hypotheses about the emergence of communication. This kind of research is performed in a variety of traditions including language evolution, developmental psychology, cognitive science, machine learning, robotics, etc. The motivations for the models are different, but the operationalizations and methods used are often similar. We identify the assumptions and explanatory targets of several most representative models and summarise the known results. We claim that some of the assumptions -such as portraying meaning in terms of mapping, focusing on the descriptive function of communication, modelling signals with amodal tokens -may hinder the success of modelling. Relaxing these assumptions and foregrounding the interactions of embodied and situated agents allows one to systematise the multiplicity of pressures under which symbolic systems evolve. In line with this perspective, we sketch the road towards modelling the emergence of meaningful symbolic communication, where symbols are simultaneously grounded in action and perception and form an abstract system.