  We discuss the birth of the non-perturbative approach to quantum gravity
known as quantum Einstein gravity, in which the gravitational interactions are
conjectured to be asymptotically safe. The interactions are assumed to be
finite and consistent at high energies thanks to a scale-invariant ultraviolet
completion. We present the framework on the basis of perturbative arguments
that originally motivated it, paying special attention to the
$\epsilon$-expansion in $d=2+\epsilon$ dimensions and the large-$N$ expansion
for $N$ the number of flavors of matter fields. The chapter is organized in
such a way that each section is mostly independent and can offer several ideas
for both conceptual and technical future developments.
