  We preform a multi-orbital analysis on the novel superconductivity in
Na_{x}CoO_{2} \cdot yH_{2}O. We construct a three-orbital model which
reproduces the band structure expected from the LDA calculation. The effective
interaction leading to the pairing is estimated by means of the perturbation
theory. It is shown that the spin triplet superconductivity is stabilized in
the wide parameter region. This is basically owing to the ferromagnetic
character of spin fluctuation. The p-wave and f-wave superconductivity are
nearly degenerate. The former is realized when the Hund's rule coupling is
large, and vice versa. In a part of the parameter space, the d-wave
superconductivity is also stabilized. We point out that the orbital degeneracy
plays an essential role for these results through the wave function of
quasi-particles. The nearly degeneracy of p-wave and f-wave superconductivity
is explained by analysing the orbital character of each Fermi surface. We
discuss the validity of some reduced models. While the single band Hubbard
model reproducing the Fermi surface is qualitatively inappropriate, we find an
effective two-orbital model appropriate for studying the superconductivity. We
investigate the vertex corrections higher than the third order on the basis of
the two-orbital model. It is shown that the vertex correction induces the
screening effect but does not affect on the qualitative results.
