Beginning with the famous Olber's paradox, paradoxes such as the missing mass, dark energy, baryon to photon ratio and cosmic zero-point energy are examined in detail. The Heisenberg-Lemaitre's units, based on the total enormous but finite mass of the Universe, are introduced and rigorous solutions of Einstein's cosmological equations for an open Universe with cosmological constant are obtained. Energy conservation after the Big Bang is consistently required.This book discusses such paradoxes in depth with physical and logical content and historical perspective, and has not too technical content in order to serve a wide audience. In the second edition, the content is updated and new sections are added.