  The theory of dependency graphs is a powerful toolbox to prove asymptotic
normality of sums of random variables. In this article, we introduce a more
general notion of weighted dependency graphs and give normality criteria in
this context. We also provide generic tools to prove that some weighted graph
is a weighted dependency graph for a given family of random variables.
  To illustrate the power of the theory, we give applications to the following
objects: uniform random pair partitions, the random graph model $G(n,M)$,
uniform random permutations, the symmetric simple exclusion process and
multilinear statistics on Markov chains. The application to random permutations
gives a bivariate extension of a functional central limit theorem of Janson and
Barbour. On Markov chains, we answer positively an open question of Bourdon and
Vall\'ee on the asymptotic normality of subword counts in random texts
generated by a Markovian source.
