  We introduce a general class of four-dimensional effective field theories
which include curved space Galileons and DBI theories possessing nonlinear
shift-like symmetries. These effective theories arise from purely gravitational
actions for 3-branes probing higher dimensional spaces. In the simplest case of
a Minkowski brane embedded in a higher dimensional Minkowski background, the
resulting four-dimensional effective field theory is the Galileon one, with its
associated Galilean symmetry and second order equations. However, much more
general structures are possible. We construct the general theory and explicitly
derive the examples obtained from embedding maximally symmetric branes in
maximally symmetric ambient spaces. Among these are Galileons and DBI theories
with second order equations that live on de Sitter or anti-de Sitter space, and
yet retain the same number of symmetries as their flat space counterparts,
symmetries which are highly non-trivial from the 4d point of view. These
theories have a rich structure, containing potentials for the scalar fields,
with masses protected by the symmetries. These models may prove relevant to the
cosmology of both the early and late universe.
