  We consider a generalisation of the Seiberg-Witten invariant to the families
Seiberg-Witten invariants of a smooth family of 4-manifolds with fibres
diffeomorphic to a 4-manifold $X$. Of particular interest is the special case
when the family has a smoothly varying K\"ahler structure. We obtain a general
computation of the invariants when $b_1=0$ in terms of characteristic classes
of some vector bundles corresponding to the cohomology groups of holomorphic
line bundles of the family. Finally, we apply the formula to some examples of
K\"ahler families where some more further explicit computations can be made. We
consider a family of $\mathbb{CP}^2$'s obtained from the projectivisation of a
rank 3 complex vector bundle, a family of $\mathbb{CP}^1\times\mathbb{CP}^1$'s
obtained as the fibre product of the projectivisation of two rank 2 complex
vector bundles and a family with fibres being the blowup of a K\"ahler surface
at a point known as the universal blowup family.
