A difficult allocation problem arises when multiple people lay claim to a scarce and indivisible good. Different allocation mechanisms produce different outcomes in terms of the goodness and fairness. Most often, a trade-off exists between good and fair outcomes. Lotteries are often considered to be a fair allocation method. John Broome developed a theory of fairness that, he argues, explains better than any other theory why this is the case. For Broome, fairness requires the proportional satisfaction of claims. Lotteries provide a contribution in fairness that sometimes outweighs the loss in goodness. This thesis provides an introduction to lotteries, an overview of Broome’s theory of fairness, and places this in the literature. Lastly, this thesis aims to defend Broome’s theory of fairness from criticism by Hooker and Lazenby. Hooker criticises Broome for using a too limited notion of fairness and argues that on some occasions proportional satisfaction of claims is unfair. I argue that Hooker’s criticisms are largely based on a misunderstanding of Broome’s position. Lazenby criticises Broome’s idea that the contribution in fairness that a lottery creates can outweigh the losses in goodness and that marginally weaker claims deserve satisfaction. I argue that his examples do not undermine Broome’s theory of fairness.