  Kontsevich's formula for rational plane curves is a recursive relation for
the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing
through $3d-1$ general points.
  We provide two proofs of this recursion: the first more direct and
combinatoric, the second more abstract.
  In order to achieve this, we introduce several moduli spaces, such as the
Deligne-Mumford-Knudsen spaces and the Kontsevich spaces, and exploit their
properties. In particular, the boundary structure of these spaces gives rise to
certain fundamental relations crucial to both proofs.
  For the second proof, we reconsider the objects in question from the
cohomological viewpoint and generalize the numbers $N_d$ to Gromov-Witten
invariants. We introduce quantum cohomology and deduce Kontsevich's formula
from the associativity of the quantum product.
  We also adapt these steps to the case of curves in
$\mathbb{P}^1\times\mathbb{P}^1$, whose bidegrees lead to slightly more
complicated but analogous results.
