I will briefly recall well-known evidence about the fact that, most probably, our Universe had an origin. I will devote some time to clarify the concept of Big Bang, namely: (i) what did Fred Hoyle mean when he said these two words for the very first time? (ii) what does it mean to the most rigorous mathematical cosmologists, e.g., the Big Bang singularity?
Does it still make sense when one involves quantum effects? A theorem of Borde, Guth and Vilenkin, which takes into account inflation and quantum vacuum fluctuations, and which extends former results of singularity theorems by Penrose, Geroch and Hawking, and a previous theorem of Borde and Vilenkin, seems to provide the last word, up to present, on the main issue here. However, the final conclusion is still dependent on the theory itself being considered in order to describe, by looking backwards, the very first stage of our expanding cosmos. Some related work of our group will be discussed.