  Building on techniques used in the case of the disc, we use a variety of
methods to develop formulae for the adjoints of composition operators on Hardy
spaces of the upper half-plane. In doing so, we prove a slight extension of a
known necessary condition for the boundedness of such operators, and use it to
provide a complete classification of the bounded composition operators with
rational symbol. We then consider some specific examples, comparing our
formulae with each other, and with other easily deduced formulae for simple
cases.
