We present a uni?ed approach to various sharp pointwise inequalities for a- lytic functions in a disk with the real part of the function on the circumference as the right-hand side. We refer to these inequalities as real-part theorems in reference to the ?rst assertion of such a kind, the celebrated Hadamard s real-part theorem (1892). The inequalities in question are frequently used in the theory of entire functions and in the analytic number theory. We hope that collecting these inequalities in one place, as well as general- ing and re?ning them, may prove useful for various applications. In particular, one can anticipate rich opportunities to extend these inequalities to analytic functions of several complex variables and solutions of partial di?erential eq- tions. The text contains revisions and extensions of recent publications of the authors 56]- 58] and some new material as well. The research of G. Kresin was supported by the KAMEA program of the Ministry of Absorption, State of Israel, and by the College of Judea and Samaria, Ariel. The work of V. Maz ya was supported by the Liverpool University and the Ohio State University. The authors record their thanks to these institutions. We are most grateful to Lev Aizenberg and Dmitry Khavinson for int- esting comments and enhancing our knowledge of the history of the topic."