In this dissertation I propose a theory of conditional probability suitable for the interpretation of probability as degree of belief. I first argue that different interpretations of probability must have different mathematical accounts, and then consider both what conditional degree of belief is, and what general form a mathematical theory of it must take. In particular, I suggest that the way to reason about the mathematical formalism is by looking at the uses of conditional probability, in particular in confirmation theory. In Chapter 5, I motivate the central problem that my account of conditional probability addresses, which is that the traditional mathematical account of conditional probability occasionally requires dividing by zero. I then consider alternate approaches to this problem, but reject them because of a series of constraints that I argue any theory of conditional degree of belief must meet. In Chapter 8 I give my positive account, according to which conditional degree of belief must be taken (at least sometimes) to be a relative notion, rather than an absolute one. Where traditional accounts say that it is a function of two propositions, I argue that it depends also on which set of alternatives to the conditioning proposition is relevant. This is a radical proposal, but I argue that in all standard uses of conditional probability, it poses no new problems, because the relevant set of alternatives is specified by the context.