Christian mission to Jews, evangelism among Jews, or proselytism to Jews, is a subset of Christian mission activity aimed specifically at Jews.
Christian proselytism soon divides into the mission of Peter "to the Jews" and the mission of Paul "to the Gentiles."
An important 2nd century source is the Dialogue with Trypho of Justin Martyr (c.140) which may be partially fictionalized, and "Trypho" may be a cypher for rabbi Tarfon but otherwise shows a level playing field and mutual respect as each participant appeals to the other.
From Constantine I, when Christianity became the official religion of the Roman empire, the position of Christians to Jews changed. Some laws were instituted which protected the rights of Jewish converts from disinheritance, other laws also protected from abuse of the privileges of conversion from those who converted from Judaism "only for a cancellation of debt;" which suggests that in some areas of the empire local incentives to conversion existed. Accounts of conversion itself are not mentioned in rabbinical sources and are not frequent in Christian sources - excepting Epiphanius of Salamis' account of the conversion of Count Joseph of Tiberias, and Sozomen's accounts of Jewish conversions in Constantinople.
During the medieval period conversions in Christian ruled lands were often conducted by force, such as in the case of the Alhambra Decree of 1492 leading to the conversos, those converted by force, and Marranos. In Muslim lands dialogue between Jews and Christians was more equal, and Jewish apologists were able to refute Christians openly. In Christian lands those such as Hasdai Crescas (c.1340–1411) could only write refutations of Christian belief at great risk.