These lecture notes are based on the course "Gravitational waves from the early universe" given at the 27th W.E. Heraeus "Saalburg" Summer School 2021 by Valerie Domcke. Ongoing and future collaborations will probe different frequency ranges of the gravitational wave spectrum, allowing for probing different stages of the early universe and Beyond Standard Model physics. Due to the very high energies involved, accelerators cannot probe them. Therefore, current knowledge about new physics is limited and relies on bounds from CMB observations and theoretical assumptions about these energy scales. While some models are in tension with CMB data, others are unconstrained in shorter wavelength scales. Nonetheless, each one of these models has a gravitational wave density spectrum that can be compared to data. These lecture notes review the formalism of gravitational waves in General Relativity and introduce stochastic gravitational waves, primordial sources, and detection efforts.