In information overload regimes, it is necessary for messages to not only provide information but also to attract attention in the first place. Bayesian surprise is an information-theoretic functional that has been experimentally shown to measure the attraction of human attention. This paper studies the limits of reliable communication under a constraint on surprise so as to limit distraction: surprise-constrained capacity. It also considers attention-seeking capacity, where the goal is to maximize both information rate and surprise to attract attention. Properties of these functions are proven. There are no nontrivial tradeoffs for surprise-constrained capacity, but an interesting tradeoff arises for attention-seeking capacity; reversing the direction of constraint does not yield essentially equivalent problems. © 2013 IEEE.