  In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a
signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any
two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the
colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ is the sign of the edge $uv$.
The substantial part of Zaslavsky's research concentrated on polynomial
invariants related to signed graph colourings rather than on the behaviour of
colourings of individual signed graphs. We continue the study of signed graph
colourings by proposing the definition of a chromatic number for signed graphs
which provides a natural extension of the chromatic number of an unsigned
graph. We establish the basic properties of this invariant, provide bounds in
terms of the chromatic number of the underlying unsigned graph, investigate the
chromatic number of signed planar graphs, and prove an extension of the
celebrated Brooks theorem to signed graphs.
