  We study the ``hyperboloidal Cauchy problem'' for linear and semi-linear wave
equations on Minkowski space-time, with initial data in weighted Sobolev spaces
allowing singular behaviour at the boundary, or with polyhomogeneous initial
data. Specifically, we consider nonlinear symmetric hyperbolic systems of a
form which includes scalar fields with a $\lambda\phi^p$ nonlinearity, as well
as wave maps, with initial data given on a hyperboloid; several of the results
proved apply to general space-times admitting conformal completions at null
infinity, as well to a large class of equations with a similar non-linearity
structure. We prove existence of solutions with controlled asymptotic
behaviour, and asymptotic expansions for solutions when the initial data have
such expansions. In particular we prove that polyhomogeneous initial data
(satisfying compatibility conditions) lead to solutions which are
polyhomogeneous at the conformal boundary Scri of the Minkowski space-time.
