To piggyback on this question a bit, can someone comment on Wilfred Madelung's work, The Succession to Mohammad, which (apparently) argues that succession to the prophet was likely hereditary. From my Google searches, I've discovered that he is a very well-respected historian and his work has even been called "magisterial". If his argument is well-sourced and convincing, and is accepted by historians of early Islamic history, it would mean that Shiite version of religious truth has more backing of secular historians contra the traditional Sunni version. In this context, I think the opinion of historians on this vexing question of Islamic history is extremely interesting to know.

Lol, this is a loaded question. There are competing historical claims from the Sunni and Shi'a traditions. I'll get into non-Muslim views in a sec, but let me explain these two:

Sunni: Muhammad did not name a successor but made it very clear that Abu Bakr should succeed him through indirect means. They point to the close relationship between the two men, the fact that Abu Bakr is alluded to in the Qur'an itself, and that Abu Bakr led the congregational prayers towards the end of Muhammad's life. This last part is particularly important because leading the prayer was done by the leader. So much so that on campaigns, the commander would lead the prayer and there's even an incident in early Islamic history books over two of Muhammad's commanders arguing about who would lead the prayer when their units were merged. Each commander assumed he was in charge and the other unit was sent as his reinforcement. The point being, leading prayer was not just an act of worship, it had a strong symbolic meaning.

Shia view: Muhammad did name a successor, and that was Ali. They also point to indirect methods by which Muhammad made it clear that Ali should succeed him. They point to an incident at a place called Ghadeer Khum. This happened after the farewell pilgrimage, where Muhammad silenced resentment towards Ali from some of the men under Ali's leadership by saying, "Whoever I am his master, Ali is also his master". They also point to the fact that Muhammad said about Ali that "You are to me as Aaron was to Moses."

By and large, non-Muslim scholarship leans towards the Sunni view in this regard, i.e. that Ali was not appointed the successor. Where non-Muslim scholarship diverges from the Sunni view and merges with the Shi'a view is that the general consensus is that Ali did expect to be the successor, or at least aspired to it. The Sunni view is that not only was Abu Bakr the successor, but Ali did not aspire to the caliphate at all until later.

The Venture of Islam by Marshall Hodgson has a pretty decent summary of the succession. I'd definitely recommend it.