What determines what we see? In contrast to the traditional "modular" understanding of perception, according to which visual processing is encapsulated from higher-level cognition, a tidal wave of recent research alleges that states such as beliefs, desires, emotions, motivations, intentions, and linguistic representations exert direct, top-down influences on what we see. There is a growing consensus that such effects are ubiquitous, and that the distinction between perception and cognition may itself be unsustainable. We argue otherwise: None of these hundreds of studies - either individually or collectively - provides compelling evidence for true top-down effects on perception, or "cognitive penetrability." In particular, and despite their variety, we suggest that these studies all fall prey to only a handful of pitfalls. And whereas abstract theoretical challenges have failed to resolve this debate in the past, our presentation of these pitfalls is empirically anchored: In each case, we show not only how certain studies could be susceptible to the pitfall (in principle), but also how several alleged top-down effects actually are explained by the pitfall (in practice). Moreover, these pitfalls are perfectly general, with each applying to dozens of other top-down effects. We conclude by extracting the lessons provided by these pitfalls into a checklist that future work could use to convincingly demonstrate top-down effects on visual perception. The discovery of substantive top-down effects of cognition on perception would revolutionize our understanding of how the mind is organized; but without addressing these pitfalls, no such empirical report will license such exciting conclusions.