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Even though one could already constrain different models in cosmology and Beyond Standard Model physics using CMB data, these models remained unconstrained at shorter wavelength scales, and knowledge of new physics at higher energy scales relied on theoretical assumptions and extrapolations to these scales. Recently, however, we have experienced the advent of gravitational-wave and multi-messenger astronomy, including the outstanding detections by the LIGO-Virgo collaboration over the past decade and the latest searches for Hellings-Downs correlations in pulsar timing data. Ongoing and future gravitational wave collaborations explore different frequency ranges of the gravitational wave spectrum. In these lecture notes, we focus on how we can probe cosmology and Beyond Standard Model physics with primordial gravitational waves. For this purpose, we review the formalism of gravitational waves in General Relativity, introduce stochastic gravitational waves, and derive the Hellings-Downs correlation for pulsar timing array searches. We comment on detection efforts and present some of the most important cosmological sources that could produce a background. Ultimately, one could compare model-dependent gravitational wave density spectra using gravitational wave data from ground-based, space-borne, and pulsar timing array searches. These lecture notes were inspired by the course "Gravitational Waves from the Early Universe" given at the 27th W.E. Heraeus "Saalburg" Summer School 2021 by Valerie Domcke.