After a brief introduction to the sixteenth and seventeenth century views of the Universe and the nineteenth century paradox of Olbers, we start the history of the cosmic expansion with Hubble's epochal discovery of the recession velocities of spiral galaxies. By then Einstein's theories of relativity were well known, but no suitable metric was known. Prior to introducing General Relativity we embark on a non-chronological derivation of the Robertson-Walker metric directly from Special Relativity and the Minkowski metric endowed with a Gaussian curvature. This permits the definition of all relativistic distance measures needed in observational astronomy. Only thereafter do we come to General Relativity, and describe some of its consequences: gravitational lensing, black holes, various tests, and the cornerstone of the standard Big Bang model, the Friedmann-Lemaitre equations. Going backwards in time towards Big Bang we first have to trace the thermal history, and then understand the needs for a cosmic inflation and its predictions. The knowledge of the Big Bang model is based notably on observations of the Cosmic Microwave Background Radiation, large scale structures, and the redshifts of distant supernovae. They tell us that gravitating matter is dominated by a dark and dissipationless component of unknown composition, and that the observable part of the Universe exhibits an accelerated expansion representing a fraction of the energy even larger than gravitating matter.