  We study the cosmology of a generalized Galileon field $\phi$ with five
covariant Lagrangians in which $\phi$ is replaced by general scalar functions
$f_{i}(\phi)$ (i=1,...,5). For these theories, the equations of motion remain
at second-order in time derivatives. We restrict the functional forms of
$f_{i}(\phi)$ from the demand to obtain de Sitter solutions responsible for
dark energy. There are two possible choices for power-law functions
$f_{i}(\phi)$, depending on whether the coupling $F(\phi)$ with the Ricci
scalar $R$ is independent of $\phi$ or depends on $\phi$. The former
corresponds to the covariant Galileon theory that respects the Galilean
symmetry in the Minkowski space-time. For generalized Galileon theories we
derive the conditions for the avoidance of ghosts and Laplacian instabilities
associated with scalar and tensor perturbations as well as the condition for
the stability of de Sitter solutions. We also carry out detailed analytic and
numerical study for the cosmological dynamics in those theories.
