  We study a free scalar field $\phi$ in a fixed curved background spacetime
subject to a higher derivative field equation of the form $F(\Box)\phi =0$,
where $F$ is a polynomial of the form $F(\Box)= \prod_i (\Box-m_i^2)$ and all
masses $m_i$ are distinct and real. Using an auxiliary field method to simplify
the calculations, we obtain expressions for the Belinfante-Rosenfeld symmetric
energy-momentum tensor and compare it with the canonical energy-momentum tensor
when the background is Minkowski spacetime. We also obtain the conserved
symplectic current necessary for quantisation and briefly discuss the issue of
negative energy versus negative norm and its relation to Reflection Positivity
in Euclidean treatments. We study, without assuming spherical symmetry, the
possible existence of finite energy static solutions of the scalar equations,
in static or stationary background geometries. Subject to various assumptions
on the potential, we establish non-existence results including a no-scalar-hair
theorem for static black holes. We consider Pais-Uhlenbeck field theories in a
cosmological de Sitter background, and show how the Hubble friction may
eliminate what would otherwise be unstable behaviour when interactions are
included.
