  We define in an axiomatic fashion a \emph{Coxeter datum} for an arbitrary
Coxeter group $W$. This Coxeter datum will specify a pair of reflection
representations of $W$ in two vector spaces linked only by a bilinear paring
without any integrality and non-degeneracy requirements. These representations
are not required to be embeddings of $W$ in the orthogonal group of any vector
space, and they give rise to a pair of inter-related root systems generalizing
the classical root systems of Coxeter groups. We obtain comparison results
between these non-orthogonal root systems and the classical root systems.
Further, we study the equivalent of the Tits cone in these non-orthogonal
representations, and we show that strong results on the geometry in the
equivalent of the Tits cone can be obtained.
