Smart nanomaterials, thanks to their stimuli-responsive properties, represent a promising class of nanostructures for biomedical applications. Their ability to undergo structural or functional changes in response to specific external cues has been exploited for the manipulation of cellular activities as a potential treatment for various health disorders. Despite their rather interesting properties and applications, most smart nanostructures are composed of inorganic materials, thus limiting their translatability in clinical applications due to potential toxicity and long-term accumulation concerns. Polydopamine nanostructures can overcome the limitations posed by inorganic smart nanomaterials thanks to their high biocompatibility, biodegradability, antioxidant effects, pH-responsiveness, tunability, surface reactivity, photothermal conversion properties, and the ability to act as photoacoustic contrast agents. In this review, we will present a general analysis of the properties of polydopamine nanostructures and of their biomedical applications. Our aim is to provide the reader with the state of the art concerning the use of polydopamine nanostructures as smart organic nanoplatforms in nanomedicine, also providing an analysis of the current limitations connected to the translation in clinical applications and the potential solutions to these challenges.