  How and why may an interacting system of many particles be described assuming
that all particles are independent and identically distributed ? This question
is at least as old as statistical mechanics itself. Its quantum version has
been rejuvenated by the birth of cold atoms physics. In particular the
experimental creation of Bose-Einstein condensates directly asks the following
variant: why and how can a large assembly of very cold interacting bosons
(quantum particles deprived of the Pauli exclusion principle) all populate the
same quantum state ? In this text I review the various mathematical techniques
allowing to prove that the lowest energy state of a bosonic system forms, in a
reasonable macroscopic limit of large particle number, a Bose-Einstein
condensate. This means that indeed in the relevant limit all particles
approximately behave as if independent and identically distributed, according
to a law determined by minimizing a non-linear Schr{\"o}dinger energy
functional. This is a particular instance of the justification of the
mean-field approximation in statistical mechanics, starting from the basic
many-body Schr{\"o}dinger Hamiltonian.
