  The chemistry of the early Universe is a fascinating field of study. Even in
the absence of any elements heavier than lithium, a surprising degree of
chemical complexity proves to be possible, giving the topic considerable
interest in its own right. In addition, the fact that molecular hydrogen plays
a key role in the formation of the first stars and galaxies means that if we
want to understand the formation of these objects, we must first develop a good
understanding of the chemical evolution of the gas. In this review, I first
give a brief introduction to the chemistry occurring in the gas prior to the
formation of the first stars and galaxies, and then go on to discuss in more
detail the main chemical processes occurring during the gravitational collapse
of gas from intergalactic to protostellar densities, and how these processes
influence the final outcome of the collapse.
