  An almost Einstein manifold satisfies equations which are a slight weakening
of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and
compactifications of certain Ricci-flat asymptotically locally Euclidean
structures are special cases. The governing equation is a conformally invariant
overdetermined PDE on a function. Away from the zeros of this the almost
Einstein structure is Einstein, while the zero set gives a scale singularity
set which may be viewed as a conformal infinity for the Einstein metric. In
this article we give a classification of the possible scale singularity spaces
and derive geometric results which explicitly relate the intrinsic conformal
geometry of these to the conformal structure of the ambient almost Einstein
manifold. Classes of examples are constructed. A compatible generalisation of
the constant scalar curvature condition is also developed. This includes almost
Einstein as a special case, and when its curvature is suitably negative, is
closely linked to the notion of an asymptotically hyperbolic structure.
