  A wonderful compactification of an orbit under the action of a semi-simple
and simply connected group is a smooth projective variety containing the orbit
as a dense open subset, and where the added boundary divisor is simple normal
crossing. We construct the wonderful compactification of the space of symmetric
and symplectic matrices, and investigate its geometry. As an application, we
describe the birational geometry of the Kontsevich spaces parametrizing conics
in Lagrangian Grassmannians.
