  We construct all the bulk and boundary unitary cubic curvature parity
invariant gravity theories in three dimensions in (anti)-de Sitter spaces. For
bulk unitarity, our construction is based on the principle that the free theory
of the cubic curvature theory reduces to one of the three known unitary
theories which are the cosmological Einstein-Hilbert theory, the quadratic
theory of the scalar curvature or the new massive gravity (NMG). Bulk and
boundary unitarity in NMG is in conflict; therefore, cubic theories that are
unitary both in the bulk and on the boundary have free theories that reduce to
the other two alternatives. We also study the unitarity of the Born-Infeld
extensions of NMG to all orders in curvature.
