Despite their small masses, comets have played an extraordinary role in enhancing our understanding of cosmic physics. It was the calculation of comet Halley's orbit and the successful prediction of its return in 1758 that firmly established the correctness of Newton's law of universal gravitation. It was the morphology of the dusty tails of comets that provided the earliest information of the nature of the interaction of solar electromagnetic radiation with dust, and it was the orientation and structure of the plasma tails of comets that led to the discovery of the solar wind. More recently, the role of the changing dusty plasma environments of comets as natural space laboratories for the study of dust-plasma interactions, and their physical and dynamical consequences, has been recognized. The forthcoming Rosetta-Philae rendezvous and lander mission will provide a unique opportunity to revisit the entire range of earlier observations of dusty plasma phenomena in a single comet, as it moves around the Sun. In this topical review, motivated by the Rosetta mission, we discuss the varying modes of interaction of the comet as it approaches the Sun, and the different dusty plasma phenomena that are expected in each case, drawing on the earlier observations, including their interpretations and prevailing open questions.