Diffuse interstellar clouds have long been thought to be relatively devoid of molecules, because of their low densities and high radiation fields. However, in the past ten years or so, a plethora of polyatomic molecules have been observed in diffuse clouds, via their rotational, vibrational, and electronic transitions. In this review, we propose a new systematic classification method for the different types of interstellar clouds: diffuse atomic, diffuse molecular, translucent, and dense. We review the observations of molecules (both diatomic and polyatomic) in diffuse clouds and discuss how molecules can be utilized as indicators of the physical and chemical conditions within these clouds. We review the progress made in the modeling of the chemistry in these clouds, and the (significant) challenges that remain in this endeavor. We also review the evidence for the existence of very large molecules in diffuse clouds, and discuss a few specific clouds of particular interest.