While direct local communication is very important for the organization of robot swarms, so far it has mostly been used for relatively simple tasks such as signaling robots preferences or states. Inspired by the emergence of meaning found in natural languages, more complex communication skills could allow robot swarms to tackle novel situations in ways that may not be a priori obvious to the experimenter. This would pave the way for the design of robot swarms with higher autonomy and adaptivity. The state of the art regarding the emergence of communication for robot swarms has mostly focused on offline evolutionary approaches, which showed that signaling and communication can emerge spontaneously even when not explicitly promoted. However, these approaches do not lead to complex, language-like communication skills, and signals are tightly linked to environmental and/or sensory-motor states that are specific to the task for which communication was evolved. To move beyond current practice, we advocate an approach to emergent communication in robot swarms based on language games. Thanks to language games, previous studies showed that cultural self-organization-rather than biological evolution-can be responsible for the complexity and expressive power of language. We suggest that swarm robotics can be an ideal test-bed to advance research on the emergence of language-like communication. The latter can be key to provide robot swarms with additional skills to support self-organization and adaptivity, enabling the design of more complex collective behaviors.