The expected utility hypothesis is an important and celebrated result which allows some sense to be made of decision-making under uncertainty. It is however a result made up of many pieces. Here an attempt is made to make as clear as possible the assumptions used in deriving the expected utility hypothesis, and so to see which ones are needed to provide a rationale for each part of the hypothesis. First we show that the additive property of expected utility follows from continuity and an independence axiom. Secondly, we argue that the probabilities used in forming an expected utility must be construed as subjective probabilities in order to have any meaning. Objective probabilities may exist, but need not coincide with subjective ones. Our third result is that concavity of the utility function that appears in an expected utility is implied by mere quasi-concavity of the overall preference ordering, provided that continuity and independence hold. This means simply that, with continuity and independence, positive risk aversion and a quasi-concave overall preference ordering are equivalent. Finally, we discuss the matter of infinite risk aversion, and find that nothing in the expected utility hypothesis requires an agent to be able to make choices between certain and uncertain events. A straightforward extension of the hypothesis to this case needs a continuity assumption not previously made, but even lexicographic preferences between certainty and uncertainty are not incompatible with expected utility maximization when only uncertain events are considered.