We will review both theoretical and obsevational aspects of gravitational wave backgrounds of cosmological origin. We will present a classifcation of backgrounds and a quantification of our ability to use them as a probe of early Universe phenomena, opening a new observational window to energy scales well above the reach of any terrestrial means. Our discussion will include the latest results from Pulsar Timing Array collaborations (e.g., Nanograv), which have recently anounced the first evidence on the existence of a background of gravitational waves. This talk wll be given in a Colloquium style, making it accessible to a wide audience of physicists.