Human–robot interaction (HRI) has escalated in notability in recent years, and multimodal communication and control strategies are necessitated to guarantee a secure, efficient, and intelligent HRI experience. In spite of the considerable focus on multimodal HRI, comprehensive disquisitions delineating various modalities and intricately analyzing their combinations remain elusive, consequently limiting holistic understanding and future advancements. This article aspires to bridge this inadequacy by conducting a profound exploration of multimodal HRI, predominantly concentrating on four principal modalities: vision, auditory and language, haptics, and physiological sensing. An extensive review encapsulating algorithmic dissection, interface devices, and applicative dimensions forms part of this discourse. This manuscript distinctively combines multimodal HRI with cognitive science, deeply probing into the three dimensions, perception, cognition, and action, thereby demystifying algorithms intrinsic to multimodal HRI. Finally, it accentuates the empirical challenges and contours preemptive trajectories for multimodal HRI in human-centric smart manufacturing.