What does it mean to say that the Universe had a beginning? There are different ways of spelling this out. I shall develop them, consider the logical relations between them, and support one as best capturing our intuitive understanding of this notion. I shall then draw a conclusion about whether Time could (it is logically possible) have a beginning. Finally I shall consider, on my preferred understanding of what it is for the Universe to have a beginning, what physical cosmology can show about whether it did in fact have a beginning.I understand by a Universe, a system of substances temporally connected to each other. I understand by a substance a thing with causal powers or liabilities, that is able to act or be acted upon. Substances will thus include both material objects and any other physical objects there may be such as chunks of energy or the fluctuating ‘vacuum’ of quantum field theory, and immaterial objects, if there are any, such as souls and ghosts. I understand by two substances being temporally connected that they exist for periods of time which are either earlier than, overlap with, or are later than each other.