The restricted three-body problem studies the dynamics of a massless
body influenced by two massive bodies, known as primaries, according to
Newton's law of gravitation. Additionally, it is assumed that the
primaries move in circular orbits around their centre of mass. Despite
centuries of study and the simplifications introduced, this problem
remains not fully understood, revealing new geometric structures and
properties. In my talk, I will present the circular restricted three-body
problem step-by-step. I will discuss the so-called low-energy dynamics
and demonstrate how contact geometry naturally emerges when examining
the three-body problem. First, I will introduce fundamental concepts
from symplectic and contact geometry. Then, I will comment on Hill's
region and Lagrange points. Finally, I will discuss what occurs just
above the low-energy level.