  We give new and rather general gluing theorems for anti-self-dual (ASD)
conformal structures, following the method suggested by Floer. The main result
is a gluing theorem for pairs of conformally ASD manifolds `joined' across a
common piece (union of connected components) of their boundaries. This theorem
genuinely operates in the b-category (in the sense of Melrose) and in general
the boundary of the joined manifold can be non-empty. The resulting metric is a
conformally ASD b-metric or, in more traditional language, a complete
conformally ASD metric with cylindrical asymptotics. We also study
hermitian-ASD conformal structures on complex surfaces in relation to
scalar-flat K\"ahler geometry. The general results are illustrated with a
simple application, showing that the blow-up of C^2 at an arbitrary finite set
of points admits scalar-flat K\"ahler metrics that are asymptotic to the
Euclidean metric at infinity. A number of vanishing theorems for the
obstruction space is also included.
