The use of robot swarms for odor source localization (OSL) can better adapt to the reality of unstable turbulence and find chemical contamination or hazard sources faster. Inspired by the collective behavior in nature, swarm intelligence (SI) is recognized as an appropriate algorithm framework for multi-robot system due to its parallelism, scalability and robustness. Applications of SI-based multi-robots for OSL problems have attracted great interest over the last two decades. In this review, we firstly summarize the trending issues in general robot OSL field through comparing some basic counterpart concepts, and then provide a detailed survey of various representative SI algorithms in multi-robot system for odor source localization. The research field originates from the first introduction of the standard particle swarm optimization (PSO) and flourishes in applying ever-increasing quantity of its variants as modified PSOs and hybrid PSOs. Moreover, other nature-inspired SI algorithms have also demonstrated the diversity and exploration of this field. The computer simulations and real-world applications reported in the literatures show that those algorithms could well solve the main problems of odor source localization but still retain the potential for further development. Lastly, we provide an outlook on possible future research directions.