The first edition of this well-known book was noted for its clear and accessible exposition of the basic theory of Hardy spaces from the concrete point of view (in the unit circle and the half plane). This second edition retains many of the features found in the first - detailed computation, an emphasis on methods - but greatly extends its coverage. The discussions of conformal mapping now include Lindelöf's second theorem and the one due to Kellogg. A simple derivation of the atomic decomposition for RH1 is given, and then used to provide an alternative proof of Fefferman's duality theorem. Two appendices by V.P. Havin have also been added: on Peter Jones' interpolation formula for RH1 and on Havin's own proof of the weak sequential completeness of L1/H1(0). Numerous other additions, emendations and corrections have been made throughout.