Our perception of persisting things is typically a perception of them as persisting. This shows up notably in our perception of things as moving (e.g. seeing a car driving past) or as being at rest (e.g. watching a coin lying in a display case): in either case, our perceptual awareness seems to be as of a single object so-and-so located over a brief duration. I here relate this aspect of perceptual experience to the current debate between so-called ‘relationist’ and ‘intentionalist’ (a.k.a. ‘representationalist’) views of perceptual experience. I argue that differences in the perception of object persistence make for the possibility of perceptual analogues of ‘Frege cases’, i.e. for contrasts in perception depending on whether or not the identity over time of a certain perceived object is perceptually manifest. Since intentionalists take perceptual experiences to have fundamentally the same structure as such paradigm intentional states as beliefs, and since a variety of theories are available to handle Frege cases for belief, the intentionalist has several options for accounting for the analogue cases in perception. Relationists, in contrast, take perceptual experiences to differ fundamentally from beliefs and intentional states, construing perception in terms of a non-intentional relation of acquaintance between subject and certain entities of which she is aware. I review a number of theoretical options for a relationist when it comes to accounting for the perception of object persistence (in general) and our perceptual Frege cases (as a special case). The most promising strategies, I suggest, invoke perceptual acquaintance with, respectively, (a) extremely fine-grained conjunctive states of affairs, or (b) states of affairs constituted by equivalence relations over temporal parts of continuants. Either strategy has drawbacks, I argue. Option (a) is hard to distinguish from a form of intentionalism. Option (b) faces a challenge over a putative difference between our perception of continuants (i.e. persisting objects) and perception of events.