It is advisable to avoid and, even better, demystify such grandiose terms as "infinity" or "singularity" in the description of the cosmos. Its proliferation does not positively contribute to the understanding of key concepts that are essential for an updated account of its origin and evolutionary history. It will be here argued that, as a matter of fact, there are no infinities in physics, in the real world: all which appear, in any given formulation of nature by means of mathematical equations, actually arise from extrapolations, which are made beyond the bounds of validity of the equations themselves. Such crucial point is rather well-known, but too often forgotten, as discussed in this paper with several examples; namely, the famous Big Bang singularity and others, which appeared before in classical mechanics and electrodynamics, and notably in the quantization of field theories. A brief description of the Universe’s history and evolution follows. Special emphasis is put on what is presently known, from detailed observations of the cosmos and, complementarily, from advanced experiments of very-high-energy physics. To conclude, a future perspective on how this knowledge might soon improve is given.