Continuously-Adaptive Discretization for Message-Passing (CAD-MP) is a new message-passing algorithm for approximate inference. Most message-passing algorithms approximate continuous probability distributions using either: a family of continuous distributions such as the exponential family; a particle-set of discrete samples; or a fixed, uniform discretization. In contrast, CAD-MP uses a discretization that is (i) non-uniform, and (ii) adaptive to the structure of the marginal distributions. Non-uniformity allows CAD-MP to localize interesting features (such as sharp peaks) in the marginal belief distributions with time complexity that scales logarithmically with precision, as opposed to uniform discretization which scales at best linearly. We give a principled method for altering the non-uniform discretization according to information-based measures. CAD-MP is shown in experiments to estimate marginal beliefs much more precisely than competing approaches for the same computational expense.