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Stars of low and intermediate mass that exhibit oscillations may show tens of detectable oscillation modes each. Oscillation modes are a powerful to constrain the internal structure and rotational dynamics of the star, hence tool allowing one to obtain an accurate stellar age. The tens of thousands of solar-like oscillators that have been discovered thus far are representative of the large diversity of fundamental stellar properties and evolutionary stages available. Because of the wide range of oscillation features that can be recognized in such stars, it is particularly challenging to properly characterize the oscillation modes in detail, especially in light of large stellar samples. Overcoming this issue requires an automated approach, which has to be fast, reliable, and flexible at the same time. In addition, this approach should not only be capable of extracting the oscillation mode properties of frequency, linewidth, and amplitude from stars in different evolutionary stages, but also able to assign a correct mode identification for each of the modes extracted. Here we present the new freely available pipeline FAMED (Fast and AutoMated pEak bagging with DIAMONDS), which is capable of performing an automated and detailed asteroseismic analysis in stars ranging from the main sequence up to the core-Helium-burning phase of stellar evolution. This, therefore, includes subgiant stars, stars evolving along the red giant branch (RGB), and stars likely evolving toward the early asymptotic giant branch. In this paper, we additionally show how FAMED can detect rotation from dipolar oscillation modes in main sequence, subgiant, low-luminosity RGB, and core-Helium-burning stars. FAMED can be downloaded from its public GitHub repository (https://github.com/EnricoCorsaro/FAMED).