The Pierre Auger Observatory, as a key actor in multi-messenger astronomy, is playing a crucial role in searching for and following-up cosmic phenomena across different channels. Data from the Observatory have been utilized for nearly 20 years to search for showers induced by Ultra-High-Energy (UHE) neutrinos with energies exceeding 0.1 EeV. Neutrino-induced showers at high zenith angles are likely to develop deep in the atmosphere, resulting in a significant electromagnetic component that distinguishes them from the cosmic-ray background. This enables the identification of candidate events from both neutrinos interacting in the atmosphere and Earth-skimming $\tau$ neutrinos. Searches have been conducted for both diffuse and point sources using data collected by the Surface Detector, a large array of over 1660 water-Cherenkov stations spread over an area of 3000 $\mathrm{km^2}$. Additionally, the Fluorescence Detector consisting of 27 telescopes has been employed to search for upward-developing air showers, as predicted by several interpretations of the 'anomalous' events detected by the ANITA detector. In this contribution, we summarize the main results obtained in these searches and discuss their astrophysical implications.