He has a track record of inflammatory speechmaking, calling for women and Christians to be barred from Egypt's presidency, denouncing Israel, and saying that the prime role of the state lies in enforcing Sharia and spreading Islam.
But he is also American-educated, an engineering professor, and committed to a pluralistic democracy. Two of his sons hold American passports. He has refused to discuss the issues that most symbolise for outsiders whether a country is to be thoroughly "Islamised" – the veil and alcohol.
The group has said it will not limit people's personal freedoms, and will respect the views of minorities.
The truth is that Mr Morsi is just one of a number of senior Brotherhood functionaries who will dictate Egypt's policies from now on, even to the extent that the army allows them to do so. He is on the Brotherhood council – and has earned his place by a seven month stretch in one of Mr Mubarak's jails in 2006 – but is outranked by its "Guide", Mohammed Badie, and a number of other leaders.
Most important of those in political terms is Khairat al-Shater, the group's chief strategist and Dr Morsi's main patron. In his religious views, Mr Shater is conservative; but he is also a multi-millionaire, English-speaking businessman whose commitment to free market principles and pragmatic social policies has done much to reassure the United States, which has held extensive meetings with the Brotherhood leadership. Mr Shater and Mr Morsi have been the public face of the Brotherhood's "Renaissance Project", the name they have given to their political campaign. There can be few observers of Egypt's economic and social crisis who could disagree that its main policies, of encouraging private enterprise and cutting corruption, promoting a more modern, technical education and fighting poverty, are not needed.
What remains to be seen is whether they really have the will, and the political and administrative skills necessary, to put them into place.