Thesis There is no self: the Buddhist and Humean bundle views are basically right
Where it stands
Hume, in the Treatise (Book 1, Part 4, Section 6), says that when he looks within he never catches "himself", only some particular perception: heat, cold, love, pain. The self is a bundle of perceptions in constant flux.
The Buddhist doctrine of anattā (non-self) goes further. In the Milindapañha, the monk Nāgasena tells King Milinda that "chariot" is just a name for the assembled parts; no part is the chariot, and there's no chariot over and above the parts. So too with persons.
Derek Parfit, in Reasons and Persons (1984), argued that personal identity is not what matters, and noted the closeness of his view to the Buddha's.
My thesis: there's no enduring self over and above physical and psychological continuity. Who will defend the self?