  We give a proof of the Khintchine inequalities in non-commutative
$L_p$-spaces for all $0< p<1$. These new inequalities are valid for the
Rademacher functions or Gaussian random variables, but also for more general
sequences, e.g. for the analogues of such random variables in free probability.
We also prove a factorization for operators from a Hilbert space to a non
commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur
maps are H\"older on semifinite von Neumann algebras. The main tool is a new
form of H\"older inequality for non commutative Lp spaces with weights.
