The movement is carried by institutions rather than by a single authority. Yeshiva University and its rabbinical school in New York, the Rabbinical Council of America as the rabbinic body, the Orthodox Union for congregations and for kashrut certification, and a network of day schools and of study programmes in Israel form the American shape of it. In Israel the corresponding world is religious Zionist: the hesder yeshivot that combine study with military service, the state religious school system, and a rabbinic tradition running through Abraham Isaac Kook.
The religious reading of the state of Israel is one of the clearest markers separating this stream from Haredi Orthodoxy. Where Haredi thought has generally been neutral or hostile towards a state established by secular means, most Modern Orthodox and religious Zionist thought reads its establishment in positive religious terms and treats service to it as a religious obligation.
The internal arguments are about direction. One is whether the community has drifted towards greater stringency and away from the confident engagement Hirsch and Soloveitchik described, a tendency often discussed under the heading of a slide to the right. Another concerns the roles open to women, where advanced Talmud study for women is now widespread and ordination is not: the Rabbinical Council of America resolved in 2015 that it would not ordain women or recognise such ordinations, and a smaller group of institutions has proceeded differently.