  This is a very comprehensible review of some key issues in Modern Cosmology.
Simple mathematical examples and analogies are always used, whenever available.
The starting point is the well know Big Bang Cosmology (BBC). We deal with the
mathematical singularities appearing in this theory and discuss some ways to
remove them. Next, and before introducing the inflationary paradigm by means of
some clear examples, we review the horizon and flatness problems of the old BBC
model. We then consider the current cosmic acceleration and, as a procedure to
deal with both periods of acceleration in a unified way, we study
quintessential inflation. Finally, the reheating stage of the universe, via
gravitational particle production, which took place after inflation ended, is
discussed in simple mathematical terms, by involving the so-called
$\alpha$-attractors in the context of quintessential inflation.
