In this paper we have endeavored to critically evaluate our present understanding of cometary atmospheres. Following a brief introduction of the significance of the study of cometary atmospheres (Section 1), the relevant photometric and spectroscopic observations are summarized in Section 2. The interaction with the solar radiation, with regard to both the excitation of the observed species as well as the dissociation of stable molecules evaporating from the nucleus, is considered in Sections 3 and 4. The gas phase chemistry likely to take place in the dense inner coma is next considered in Section 5. The exospheric and hydrodynamic models of the expanding cometary atmosphere are considered in detail in Section 6, and both their limitations as well as possible improvements are discussed. The observed chemical composition of the neutral atmosphere and the inferred chemical composition of the volatile component of the nucleus, together with possible variations between different classes of comets is next considered in Section 7, and their possible cosmogonic significance is discussed. In conclusion, some of the important directions in which future research should progress, in order to provide more complete and secure knowledge of cometary atmospheres, are stressed (Section 8).