  In this brief article, I summarize attempts with collaborators over the last
couple of years to extend the Galileon idea in two important ways. I discuss
the effective field theory construction arising from co-dimension greater than
one flat branes embedded in a flat background - the multi-Galileons - and then
describe symmetric covariant versions of the Galileons, more suitable for
general cosmological applications. These generalized Galileons can be thought
of as interesting four-dimensional field theories in their own rights, but the
work described here may also make it easier to embed them into higher
dimensional theories. I also briefly mention some intriguing properties,
including freedom from ghosts and a non-renormalization theorem, that hint at
possible applications in particle physics and cosmology
