  This paper presents new classes of exact radial solutions to the nonlinear
ordinary differential equation that arises as a saddle-point condition for a
Euclidean scalar field theory in $D$-dimensional spacetime. These solutions are
found by exploiting the dimensional consistency of the radial differential
equation for a single {\it massless} scalar field, which allows one to
transform to an autonomous equation. For massive theories the radial equation
is not exactly solvable but the massless solutions provide useful
approximations to the results for the massive case. The solutions presented
here depend on the power of the interaction and on the spatial dimension, both
of which may be noninteger. Scalar equations arising in the study of conformal
invariance fit into this framework and classes of new solutions are found.
These solutions exhibit two distinct behaviours as $D\to2$ from above.
