  In this thesis, we generalize the Koszul duality for associative algebras and
operads to PROPs. The operads are algebraic objects that represent the
operations with multiple inputs but only one output acting on a certain type of
algebras. A PROP models the operations with multiple inputs and multiple
outputs acting on algebraic structures like bialgebras and Lie bialgebras. We
also give a construction of the free monoid (for a monoidal product
non-necessarily bilinear). We generalize the Poincare series of operads to
PROPs and we prove a functional equation for Poincare series of Koszul PROPs.
