  We consider the factorisation problem for bialgebras: when a bialgebra $K$
factorises as $K=HL$, where $H$ and $L$ are algebras and coalgebras (but not
necessarly bialgebras). Given two maps $R: H\ot L\to L\ot H$ and $W:L\ot H\to
H\ot L$, we introduce a product $L_W\bowtie_R H$ and give necessary and
sufficient conditions for $L_W\bowtie_R H$ to be a bialgebra. It turns out that
$K$ factorises as $K=HL$ if and only if $K\cong L_W\bowtie_R H$ for some maps
$R$ and $W$. As examples of this product we recover constructions introduced by
Majid and Radford. Also, some of the pointed Hopf algebras that were recently
constructed bu Beattie, D\u asc\u alescu and Gr\"unenfelder appear as special
cases.
