Some of the most compelling applications of online convex optimization, including online prediction and classification, are unconstrained: the natural feasible set is ℝn. Existing algorithms fail to achieve sub-linear regret in this setting unless constraints on the comparator point ẋ are known in advance. We present algorithms that, without such prior knowledge, offer near-optimal regret bounds with respect to any choice of ẋ. In particular, regret with respect to ẋ = 0 is constant. We then prove lower bounds showing that our guarantees are near-optimal in this setting.