The Pierre Auger Observatory is the largest air-shower experiment in the world, offering an unprecedented exposure not only to ultra-high-energy (UHE, $E > 10^{17}$ eV}) cosmic rays, but also to UHE neutral particles, specifically photons and neutrinos. Since the beginning of data taking almost 20 years ago, a number of searches for UHE photons and neutrinos using the different detector systems of the Observatory have been carried out. These searches led to some of the most stringent upper limits on the diffuse—i.e., direction-independent, unresolved—fluxes of photons and neutrinos in the UHE regime. These limits severely constrain current models for the origin of UHE cosmic rays and underline the capabilities of the Pierre Auger Observatory and its leading role in the context of multimessenger astronomy at the highest energies. In this contribution, we give an overview of the current activities concerning searches for UHE photons and neutrinos in the data from the Pierre Auger Observatory. The latest results of the searches for diffuse fluxes of photons and neutrinos will be shown. Furthermore, the follow- up searches for UHE photons and neutrinos in association with transient events, such as gravitational wave events, will be summarized. In addition, future perspectives in view of the ongoing $\textit{AugerPrime}$ detector upgrade will be discussed, which will further improve the sensitivity of the Pierre Auger Observatory to neutral particles at the highest energies.